Page No 17.10:
Question 1:
Prove that the function f(x) = loge x is increasing on (0, ∞).
Answer:
Page No 17.10:
Question 2:
Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1.
Answer:
Page No 17.10:
Question 3:
Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R.
Answer:
Page No 17.10:
Question 4:
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R.
Answer:
Page No 17.10:
Question 5:
Show that f(x) = is a decreasing function on (0, ∞).
Answer:
Page No 17.10:
Question 6:
Show that f(x) = decreases in the interval [0, ∞) and increases in the interval (−∞, 0].
Answer:
Page No 17.10:
Question 7:
Show that f(x) = is neither increasing nor decreasing on R.
Answer:
Page No 17.10:
Question 8:
Without using the derivative, show that the function f (x) = | x | is
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0).
Answer:
Page No 17.10:
Question 9:
Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R.
Answer:
Page No 17.33:
Question 1:
Find the intervals in which the following functions are increasing or decreasing.
(i) f(x) = 10 − 6x − 2x2
(ii) f(x) = x2 + 2x − 5
(iii) f(x) = 6 − 9x − x2
(iv) f(x) = 2x3 − 12x2 + 18x + 15
(v) f(x) = 5 + 36x + 3x2 − 2x3
(vi) f(x) = 8 + 36x + 3x2 − 2x3
(vii) f(x) = 5x3 − 15x2 − 120x + 3
(viii) f(x) = x3 − 6x2 − 36x + 2
(ix) f(x) = 2x3 − 15x2 + 36x + 1
(x) f(x) = 2x3 + 9x2 + 12x + 20
(xi) f(x) = 2x3 − 9x2 + 12x − 5
(xii) f(x) = 6 + 12x + 3x2 − 2x3
(xiii) f(x) = 2x3 − 24x + 107
(xiv) f(x) = −2x3 − 9x2 − 12x + 1
(xv) f(x) = (x − 1) (x − 2)2
(xvi) f(x) = x3 − 12x2 + 36x + 17
(xvii) f(x) = 2x3 − 24x + 7
(xviii)
(xix) f(x) = x4 − 4x
(xx)
(xxi) f(x) = x4 − 4x3 + 4x2 + 15
(xxii) f(x) = , x > 0
(xxiii) f(x) = x8 + 6x2
(xxiv) f(x) = x3 − 6x2 + 9x + 15
(xxv)
(xxvi)
(xxvii)
(xxviii)
(xxix)
Answer:
(xxix)
Thus, for the increasing function the interval is and for the decreasing function .
Page No 17.34:
Question 2:
Determine the values of x for which the function f(x) = x2 − 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x2 − 6x + 9 where the normal is parallel to the line y = x + 5.
Answer:
Let (x, y) be the coordinates on the given curve where the normal to the curve is parallel to the given line.
Slope of the given line = 1
Page No 17.34:
Question 3:
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing.
Answer:
Page No 17.34:
Question 4:
Show that f(x) = e2x is increasing on R.
Answer:
Page No 17.34:
Question 5:
Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0.
Answer:
Page No 17.34:
Question 6:
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0.
Answer:
Page No 17.34:
Question 7:
Show that f(x) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π).
Answer:
Page No 17.34:
Question 8:
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π).
Answer:
Page No 17.34:
Question 9:
Show that f(x) = x − sin x is increasing for all x ∈ R.
Answer:
Page No 17.34:
Question 10:
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R.
Answer:
Page No 17.34:
Question 11:
Show that f(x) = cos2 x is a decreasing function on (0, π/2).
Answer:
Page No 17.34:
Question 12:
Show that f(x) = sin x is an increasing function on (−π/2, π/2).
Answer:
Page No 17.34:
Question 13:
Show that f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π).
Answer:
Page No 17.34:
Question 14:
Show that f(x) = tan x is an increasing function on (−π/2, π/2).
Answer:
Page No 17.34:
Question 15:
Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2).
Answer:
Page No 17.34:
Question 16:
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8).
Answer:
Page No 17.34:
Question 17:
Show that the function f(x) = cotl(sinx + cosx) is decreasing on and increasing on .
Answer:
Page No 17.34:
Question 18:
Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0.
Answer:
Page No 17.34:
Question 19:
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1).
Answer:
Page No 17.34:
Question 20:
Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R.
Answer:
Page No 17.35:
Question 21:
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R.
Answer:
Page No 17.35:
Question 22:
State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 − 6x + 3 is increasing on the interval [4, 6].
Answer:
Page No 17.35:
Question 23:
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4).
Answer:
Page No 17.35:
Question 24:
Show that f(x) = tan−1 x − x is a decreasing function on R.
Answer:
Page No 17.35:
Question 25:
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3).
Answer:
Page No 17.35:
Question 26:
Find the intervals in which f(x) = log (1 + x) − is increasing or decreasing.
Answer:
Page No 17.35:
Question 27:
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing.
Answer:
Page No 17.35:
Question 28:
Show that the function f given by f(x) = 10x is increasing for all x.
Answer:
Page No 17.35:
Question 29:
Prove that the function f given by f(x) = x − [x] is increasing in (0, 1).
Answer:
Page No 17.35:
Question 30:
Prove that the following functions are increasing on R.
(i) f3 + 40 + 240
(ii)
Answer:
(i)
So, f(x) is increasing on R.
(ii)
So, f(x) is increasing on R.
Page No 17.35:
Question 31:
Prove that the function f given by f(x) = log cos x is strictly increasing on (−π/2, 0) and strictly decreasing on (0, π/2).
Answer:
Page No 17.35:
Question 32:
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R.
Answer:
Page No 17.35:
Question 33:
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π)
Answer:
Page No 17.35:
Question 34:
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2).
Answer:
Page No 17.35:
Question 35:
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R.
Answer:
Page No 17.35:
Question 36:
Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R.
Answer:
Page No 17.35:
Question 37:
Show that f(x) = x + cos x − a is an increasing function on R for all values of a.
Answer:
Page No 17.35:
Question 38:
Let f defined on [0, 1] be twice differentiable such that | f"(x) | ≤ 1 for all x ∈ [0, 1]. If f(0) = f(1), then show that | f'(x) | < 1 for all x ∈ [ 0, 1].
Answer:
If a function is continuous and differentiable and f(0) = f(1) in given domain x ∈ [0, 1],
then by Rolle's Theorem;
f'(x) = 0 for some x ∈ [0, 1]
Given: |f"(x)| ≤ 1
On integrating both sides we get,
|f'(x)| ≤ x
Now, within interval x ∈ [0, 1]
We get, |f' (x)| < 1.
Page No 17.35:
Question 39:
Find the intervals in which f(x) is increasing or decreasing:
(i) f(x) = x|x|, x R
(ii) f(x) = sinx + |sinx|, 0 < x
(iii) f(x) = sinx(1 + cosx), 0 < x <
[CBSE 2014]
Answer:
Page No 17.39:
Question 1:
What are the values of 'a' for which f(x) = ax is increasing on R?
Answer:
Page No 17.39:
Question 2:
What are the values of 'a' for which f(x) = ax is decreasing on R?
Answer:
Page No 17.39:
Question 3:
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain.
Answer:
Page No 17.39:
Question 4:
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain.
Answer:
Page No 17.39:
Question 5:
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R.
Answer:
Page No 17.39:
Question 6:
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R.
Answer:
Page No 17.39:
Question 7:
Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R.
Answer:
Page No 17.40:
Question 8:
Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R.
Answer:
Page No 17.40:
Question 9:
Write the set of values of k for which f(x) = kx − sin x is increasing on R.
Answer:
Page No 17.40:
Question 10:
If g (x) is a decreasing function on R and f(x) = tan−1 [g (x)]. State whether f(x) is increasing or decreasing on R.
Answer:
Page No 17.40:
Question 11:
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R.
Answer:
Page No 17.40:
Question 12:
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing.
Answer:
Page No 17.40:
Question 13:
State whether f(x) = tan x − x is increasing or decreasing its domain.
Answer:
Page No 17.40:
Question 14:
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R.
Answer:
Page No 17.40:
Question 1:
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
(a) (0, ∞)
(b) (−∞, 0)
(c) (1, ∞)
(d) (−∞, 1)
Answer:
(b) (−∞, 0)
Page No 17.40:
Question 2:
The function f(x) = cot−1 x + x increases in the interval
(a) (1, ∞)
(b) (−1, ∞)
(c) (−∞, ∞)
(d) (0, ∞)
Answer:
(c) (−∞, ∞)
Page No 17.40:
Question 3:
The function f(x) = xx decreases on the interval
(a) (0, e)
(b) (0, 1)
(c) (0, 1/e)
(d) none of these
Answer:
(c) (0, 1/e)
Page No 17.40:
Question 4:
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
(a) (1, 2)
(b) (2, 3)
(c) (1, 3)
(d) (2, 4)
Answer:
(b) (2, 3)
Page No 17.40:
Question 5:
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
(a) (−∞, 4)
(b) (4, ∞)
(c) (−∞, 8)
(d) (8, ∞)
Answer:
(a) (−∞, 4)
Page No 17.40:
Question 6:
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy
(a) a2 − 3b − 15 > 0
(b) a2 − 3b + 15 > 0
(c) a2 − 3b + 15 < 0
(d) a > 0 and b > 0
Answer:
(c) a2 − 3b + 15 < 0
Page No 17.40:
Question 7:
The function is of the following types:
(a) even and increasing
(b) odd and increasing
(c) even and decreasing
(d) odd and decreasing
Answer:
(b) odd and increasing
Page No 17.40:
Question 8:
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
(a) a ∈ (1/2, ∞)
(b) a ∈ (−1/2, 1/2)
(c) a = 1/2
(d) a ∈ R
Answer:
Page No 17.40:
Question 9:
Let where g (x) is monotonically increasing for 0 < x < Then, f(x) is
(a) increasing on (0, π/2)
(b) decreasing on (0, π/2)
(c) increasing on (0, π/4) and decreasing on (π/4, π/2)
(d) none of these
Answer:
(a) increasing on (0, /2)
Page No 17.40:
Question 10:
Let f(x) = x3 − 6x2 + 15x + 3. Then,
(a) f(x) > 0 for all x ∈ R
(b) f(x) > f(x + 1) for all x ∈ R
(c) f(x) is invertible
(d) none of these
Answer:
(c) f(x) is invertible
f(x) =x3 − 6x2 + 15x + 3
Page No 17.41:
Question 11:
The function f(x) = x2 e−x is monotonic increasing when
(a) x ∈ R − [0, 2]
(b) 0 < x < 2
(c) 2 < x < ∞
(d) x < 0
Answer:
(b) 0 < x < 2
Page No 17.41:
Question 12:
Function f(x) = cos x − 2 λ x is monotonic decreasing when
(a) λ > 1/2
(b) λ < 1/2
(c) λ < 2
(d) λ > 2
Answer:
(a) λ > 1/2
Page No 17.41:
Question 13:
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
(a) monotonically increasing
(b) monotonically decreasing
(c) not monotonic
(d) constant
Answer:
(b) monotonically decreasing
Page No 17.41:
Question 14:
Function f(x) = x3 − 27x + 5 is monotonically increasing when
(a) x < −3
(b) | x | > 3
(c) x ≤ −3
(d) | x | ≥ 3
Answer:
(d) | x | ≥ 3
Page No 17.41:
Question 15:
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
(a) x < 2
(b) x > 2
(c) x > 3
(d) 1 < x < 2
Answer:
(d) 1 < x < 2
Page No 17.41:
Question 16:
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
(a) k < 3
(b) k ≤ 3
(c) k > 3
(d) k ≥ 3
Answer:
(c) k > 3
Page No 17.41:
Question 17:
f(x) = 2x − tan−1 x − log is monotonically increasing when
(a) x > 0
(b) x < 0
(c) x ∈ R
(d) x ∈ R − {0}
Answer:
(c) x ∈ R
Page No 17.41:
Question 18:
Function f(x) = | x | − | x − 1 | is monotonically increasing when
(a) x < 0
(b) x > 1
(c) x < 1
(d) 0 < x < 1
Answer:
(d) 0 < x < 1
Page No 17.41:
Question 19:
Every invertible function is
(a) monotonic function
(b) constant function
(c) identity function
(d) not necessarily monotonic function
Answer:
(a) monotonic function
We know that "every invertible function is a monotonic function".
Page No 17.41:
Question 20:
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
(a) increasing
(b) decreasing
(c) constant
(d) none of these
Answer:
(b) decreasing
Page No 17.41:
Question 21:
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
(a) a = b
(b)
(c)
(d)
Answer:
(c)
Page No 17.41:
Question 22:
The function
(a) strictly increasing
(b) strictly decreasing
(c) neither increasing nor decreasing
(d) none of these
Answer:
(a) strictly increasing
Page No 17.41:
Question 23:
The function is increasing, if
(a) λ < 1
(b) λ > 1
(c) λ < 2
(d) λ > 2
Answer:
(d) λ > 2
Page No 17.41:
Question 24:
Function f(x) = ax is increasing on R, if
(a) a > 0
(b) a < 0
(c) 0 < a < 1
(d) a > 1
Answer:
(d) a > 1
Page No 17.41:
Question 25:
Function f(x) = loga x is increasing on R, if
(a) 0 < a < 1
(b) a > 1
(c) a < 1
(d) a > 0
Answer:
(b) a > 1
Page No 17.41:
Question 26:
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
(a) increases on [0, a]
(b) decreases on [0, a]
(c) increases on [−a, 0]
(d) decreases on [a, 2a]
Answer:
Given: ϕ(x) = f(x) + f(2a − x)
Differentiating above equation with respect to x we get,
ϕ'(x) = f'(x) − f(2a − x) .....(1)
Since, f''(x) > 0, f'(x) is an increasing function.
Now,
when
Considering equation (1) and (2) we get,
ϕ'(x) ≤ 0
⇒ ϕ'(x) is decreasing in [0, a]
Page No 17.41:
Question 27:
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
(a) k ∈ (2, ∞)
(b) k ∈ (−∞, 2)
(c) k ∈ (4, ∞)
(d) k ∈ (−∞, 4).
Answer:
(d) k ∈ (−∞, 4)
Page No 17.41:
Question 28:
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
(a) increasing
(b) decreasing
(c) constant
(d) none of these
Answer:
Hence, the given function is increasing .
Page No 17.42:
Question 29:
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
(a) −1 ≤ k < 1
(b) k < −1 or k > 1
(c) 0 < k < 1
(d) −1 < k < 0
Answer:
(a)
Page No 17.42:
Question 30:
The function f(x) = x9 + 3x7 + 64 is increasing on
(a) R
(b) (−∞, 0)
(c) (0, ∞)
(d) R0
Answer:
(a) R
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