RD Sharma solutions for Mathematics [English] Class 12 chapter 17 - Increasing and Decreasing Functions [Latest edition]

Prove that the function f(x) = log_a x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?

Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?

Show that f(x) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?

Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?

Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .

Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R ?

Find the interval in which the following function are increasing or decreasing f(x) = 10 − 6x − 2x²  ?

Find the interval in which the following function are increasing or decreasing  f(x) = x² + 2x − 5  ?

Find the interval in which the following function are increasing or decreasing  f(x) = 6 − 9x − x²  ?

Find the interval in which the following function are increasing or decreasing   f(x) = 2x³ − 12x² + 18x + 15 ?

Find the interval in which the following function are increasing or decreasing f(x) = 5 + 36x + 3x² − 2x³ ?

Find the interval in which the following function are increasing or decreasing f(x) = 8 + 36x + 3x² − 2x³ ?

Find the interval in which the following function are increasing or decreasing  f(x) = 5x³ − 15x² − 120x + 3 ?

Find the interval in which the following function are increasing or decreasing f(x) = x³ − 6x² − 36x + 2 ?

Find the interval in which the following function are increasing or decreasing f(x) = 2x³ − 15x² + 36x + 1 ?

Find the interval in which the following function are increasing or decreasing f(x) = 2x³ + 9x² + 12x + 20  ?

Find the interval in which the following function are increasing or decreasing f(x) = 2x³ − 9x² + 12x − 5 ?

Find the interval in which the following function are increasing or decreasing f(x) = 6 + 12x + 3x² − 2x³ ?

Find the interval in which the following function are increasing or decreasing  f(x) = 2x³ − 24x + 107  ?

Find the interval in which the following function are increasing or decreasing f(x) = −2x³ − 9x² − 12x + 1  ?

Find the interval in which the following function are increasing or decreasing f(x) = (x − 1) (x − 2)² ?

Find the interval in which the following function are increasing or decreasing f(x) = x³ − 12x² + 36x + 17 ?

Find the interval in which the following function are increasing or decreasing  f(x) = 2x³ − 24x + 7 ?

Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\] ?

Find the interval in which the following function are increasing or decreasing f(x) = x⁴ − 4x ?

Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?

Find the interval in which the following function are increasing or decreasing  f(x) = x⁴ − 4x³ + 4x² + 15 ?

Find the interval in which the following function are increasing or decreasing  f(x) =  \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\]  x > 0 ?

Find the interval in which the following function are increasing or decreasing f(x) = x⁸ + 6x²  ?

Find the interval in which the following function are increasing or decreasing f(x) = x³ − 6x² + 9x + 15 ?

Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?

Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5\] ?

Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?

Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?

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 ? 

Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?

Show that f(x) = e^2x is increasing on R.

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) ?

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).

Show that the function f(x) = cot \[-\] ^l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?

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] ?

Find the intervals in which f(x) = log (1 + x) −\[\frac{x}{1 + x}\] is increasing or decreasing ?

Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?

Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?

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) ?

Prove that the function f given by f(x) = x³ − 3x² + 4x is strictly increasing on R ?

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).

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] ?

Find the interval in which f(x) is increasing or decreasing f(x) = x|x|, x \[\in\] R ?

Find the interval in which f(x) is increasing or decreasing f(x) = sinx + |sin x|, 0 < x \[\leq 2\pi\] ?

Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?

The interval of increase of the function f(x) = x − e^x + tan (2π/7) is

The function f(x) = cot⁻¹ x + x increases in the interval

The function f(x) = x^x decreases on the interval

The function f(x) = 2 log (x − 2) − x² + 4x + 1 increases on the interval

If the function f(x) = 2x² − kx + 5 is increasing on [1, 2], then k lies in the interval

Let f(x) = x³ + ax² + bx + 5 sin²x be an increasing function on the set R. Then, a and b satisfy.

The function \[f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\] is of the following types:

If the function f(x) = 2 tan x + (2a + 1) log_e | sec x | + (a − 2) x is increasing on R, then

Let \[f\left( x \right) = \tan^{- 1} \left( g\left( x \right) \right),\],where g (x) is monotonically increasing for 0 < x < \[\frac{\pi}{2} .\] Then, f(x) is

Let f(x) = x³ − 6x² + 15x + 3. Then,

The function f(x) = x² e^−x is monotonic increasing when

Function f(x) = cos x − 2 λ x is monotonic decreasing when

In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is

Function f(x) = x³ − 27x + 5 is monotonically increasing when

Function f(x) = 2x³ − 9x² + 12x + 29 is monotonically decreasing when

If the function f(x) = kx³ − 9x² + 9x + 3 is monotonically increasing in every interval, then

f(x) = 2x − tan⁻¹ x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when

Function f(x) = | x | − | x − 1 | is monotonically increasing when

Every invertible function is

In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is

If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then

The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is 

The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if

Function f(x) = a^x is increasing on R, if

Function f(x) = log_a x is increasing on R, if

Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)

If the function f(x) = x² − kx + 5 is increasing on [2, 4], then

The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is

If the function f(x) = x³ − 9kx² + 27x + 30 is increasing on R, then

The function f(x) = x⁹ + 3x⁷ + 64 is increasing on