Prove that the function f(x) = log_e x is increasing on (0, ∞).

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

Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R.

Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R.

Show that is a decreasing function on (0, ∞).

Show that decreases in the interval [0, ∞) and increases in the interval (-∞, 0].

Show that 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 intervals in which the following functions are increasing or decreasing.

f(x) = 10 – 6x – 2x²

Find the intervals in which the following functions are increasing or decreasing.

f(x) = x² + 2x – 5

Find the intervals in which the following functions are increasing or decreasing.

f(x) = 6 – 9x – x²

Find the intervals in which the following functions are increasing or decreasing.

f(x) = 2x³ – 12x² + 18x + 15

Find the intervals in which the following functions are increasing or decreasing.

f(x) = 5 + 36x + 3x² – 2x³

Find the intervals in which the following functions are increasing or decreasing.

f(x) = 8 + 36x + 3x² – 2x³

Find the intervals in which the following functions are increasing or decreasing.

f(x) = 5x³ – 15x² – 120x + 3

Find the intervals in which the following functions are increasing or decreasing.

f(x) = x³ – 6x² – 36x + 2

Find the intervals in which the following functions are increasing or decreasing.

f(x) = 2x³ – 15x² + 36x + 1

Find the intervals in which the following functions are increasing or decreasing.

f(x) = 2x³ + 9x² + 12x + 20

Find the intervals in which the following functions are increasing or decreasing.

f(x) = 2x³ – 9x² + 12x – 5

Find the intervals in which the following functions are increasing or decreasing.

f(x) = 6 + 12x + 3x² – 2x³

Find the intervals in which the following functions are increasing or decreasing.

f(x) = 2x³ – 24x + 107

Find the intervals in which the following functions are increasing or decreasing.

f(x) = – 2x³ – 9x² – 12x + 1

Find the intervals in which the following functions are increasing or decreasing.

f(x) = (x – 1) (x – 2)²

Find the intervals in which the following functions are increasing or decreasing.

f(x) = x³ – 12x² + 36x + 17

Find the intervals in which the following functions are increasing or decreasing.

f(x) = 2x³ – 24x + 7

Find the intervals in which the following functions are increasing or decreasing.

Find the intervals in which the following functions are increasing or decreasing.

f(x) = x⁴ – 4x

Find the intervals in which the following functions are increasing or decreasing.

Find the intervals in which the following functions are increasing or decreasing.

f(x) = x⁴ – 4x³ + 4x² + 15

Find the intervals in which the following functions are increasing or decreasing.

Find the intervals in which the following functions are increasing or decreasing.

f(x) = x⁸ + 6x²

Find the intervals in which the following functions are increasing or decreasing.

f(x) = x³ – 6x² + 9x + 15

Find the intervals in which the following functions are increasing or decreasing.

f(x) = {x (x – 2)}²

Find the intervals in which the following functions are increasing or decreasing.

f(x) = 3x⁴ – 4x³ – 12x² + 5

Find the intervals in which the following functions are increasing or decreasing.

Find the intervals in which the following functions are increasing or decreasing.

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 , x ≠ 0 is a decreasing function for all x ≠ 0.

Show that f(x) = log_a x, 0 < a < 1 is a decreasing function for all x > 0.

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) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π).

Show that f(x) = x – sin x is increasing for all x ϵ R.

Show that f(x) = x³ – 15x² + 75x – 50 is an increasing function for all x ϵ R.

Show that f(x) = cos² x is a decreasing function on (0, π/2).

Show that f(x) = sin x is an increasing function on (–π/2, π/2).

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 f(x) = tan x is an increasing function on (–π/2, π/2).

Show that f(x) = tan^–1 (sin x + cos x) is a decreasing function on the interval (π/4, π /2).

Show that the function is decreasing on

Show that the function f(x) = cot^–1 (sin x + cos x) is decreasing on (0, π/4) and increasing on (π/4, π/2).

Show that f(x) = (x – 1) e^x + 1 is an increasing function for all x > 0.

Show that the function x² – x + 1 is neither increasing nor decreasing on (0, 1).

Show that f(x) = x⁹ + 4x⁷ + 11 is an increasing function for all x ϵ R.

Prove that the function f(x) = x³ – 6x² + 12x – 18 is increasing on R.

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].

Show that f(x) = sin x – cos x is an increasing function on (–π /4, π /4)?

Show that f(x) = tan^–1 x – x is a decreasing function on R ?

Determine whether f(x) = x/2 + sin x is increasing or decreasing on (–π /3, π/3) ?

Find the intervals in which f(x) = (x + 2)e^–x is increasing or decreasing ?

Show that the function f given by f(x) = 10^x is increasing for all x ?

Prove that the function f given by f(x) = x – [x] is increasing in (0, 1) ?

Prove that the following function is increasing on r?

i. f(x) = 3x⁵ + 40x³ + 240x

ii. f(x) = 4x³ – 18x² + 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 ?

33 Prove that the function f(x) = cos x is :

i. strictly decreasing on (0, π)

ii. strictly increasing in (π, 2π)

iii. neither increasing nor decreasing in (0, 2 π)

Show that f(x) = – x sin x is an increasing function on (0, π/2) ?

Find the value(s) of a for which f(x) = – ax is an increasing function on R ?

Find the values of b for which the function f(x) = sin x – bx + c is a decreasing function on R ?

Show that f(x) = x + cos x – a is an increasing function on R for all values of a ?

Let F defined on [0, 1] be twice differentiable such that | f”(x) ≤ 1 for all x ϵ [0, 1]. If f(0) = f(1), then show that |f’(x) | < 1 for all x ϵ [0, 1] ?

Find the intervals in which f(x) is increasing or decreasing :

i. f(x) = x |x|, x ϵ R

ii. f(x) = sin x + |sin x|, 0 < x ≤ 2 π

iii. f(x) = sin x (1 + cos x), 0 < x < π/2

Mark the correct alternative in the following:

Mark the correct alternative in the following:

The function f(x) = cos^–1 x + x increases in the interval.

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

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

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

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

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The function is of the following types:

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

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Let f(x) = tan^–1 (g(x)), where g(x) is monotonically increasing for Then, f(x) is

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Let f(x) = x³ – 6x² + 15x + 3. Then,

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

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Function f(x) = cosx – 2λ x is monotonic decreasing when

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

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

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

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

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f(x) = 2x – tan^–1 x – logis monotonically increasing when

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

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Every invertible function is

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

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

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The function is

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The function is increasing, if

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

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

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Let ϕ(x) = f(x) + f(2a – x) and f’’(x) > 0 for all xϵ[0, a]. The, ϕ(x)

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

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The function defined on is

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If the function f(x) = x³ – 9k x² + 27x + 30 is increasing on R, then

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