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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 – 2x^{2}

f(x) = x^{2} + 2x – 5

f(x) = 6 – 9x – x^{2}

f(x) = 2x^{3} – 12x^{2} + 18x + 15

f(x) = 5 + 36x + 3x^{2} – 2x^{3}

f(x) = 8 + 36x + 3x^{2} – 2x^{3}

f(x) = 5x^{3} – 15x^{2} – 120x + 3

f(x) = x^{3} – 6x^{2} – 36x + 2

f(x) = 2x^{3} – 15x^{2} + 36x + 1

f(x) = 2x^{3} + 9x^{2} + 12x + 20

f(x) = 2x^{3} – 9x^{2} + 12x – 5

f(x) = 6 + 12x + 3x^{2} – 2x^{3}

f(x) = 2x^{3} – 24x + 107

f(x) = – 2x^{3} – 9x^{2} – 12x + 1

f(x) = (x – 1) (x – 2)^{2}

f(x) = x^{3} – 12x^{2} + 36x + 17

f(x) = 2x^{3} – 24x + 7

f(x) = x^{4} – 4x

f(x) = x^{4} – 4x^{3} + 4x^{2} + 15

f(x) = x^{8} + 6x^{2}

f(x) = x^{3} – 6x^{2} + 9x + 15

f(x) = {x (x – 2)}^{2}

f(x) = 3x^{4} – 4x^{3} – 12x^{2} + 5

Determine the values of x for which the function f(x) = x^{2} – 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x^{2} – 6x + 9 where the normal is parallel to the liney = 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) = x^{3} – 15x^{2} + 75x – 50 is an increasing function for all x ϵ R.

Show that f(x) = cos^{2} 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 x^{2} – x + 1 is neither increasing nor decreasing on (0, 1).

Show that f(x) = x^{9} + 4x^{7} + 11 is an increasing function for all x ϵ R.

Prove that the function f(x) = x^{3} – 6x^{2} + 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) = x^{2} – 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 interval in which is increasing or decreasing ?

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) = 3x^{5} + 40x^{3} + 240x

ii. f(x) = 4x^{3} – 18x^{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) = x^{3} – 3x^{2} + 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:

The interval of increase of the function f(x) = x – e^{x} + tan is

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

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

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

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

Let f(x) = x^{3} + ax^{2} + bx + 5 sin^{2}x be an increasing function on the set R. Then, a and b satisfy.

The function is of the following types:

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

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

Let f(x) = x^{3} – 6x^{2} + 15x + 3. Then,

The function f(x) = x^{2} e^{-x} is monotonic increasing when

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

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

Function f(x) = x^{3}– 27x +5 is monotonically increasing when

Function f(x) = 2x^{3} – 9x^{2} + 12x + 29 is monotonically decreasing when

If the function f(x) = kx^{3} – 9x^{2} + 9x + 3 is monotonically increasing in every interval, then

f(x) = 2x – tan^{–1} x – logis 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 is

The function is increasing, if

Function f(x) = a^{x} is increasing or 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 xϵ[0, a]. The, ϕ(x)

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

The function defined on is

If the function f(x) = x^{3} – 9k x^{2} + 27x + 30 is increasing on R, then

The function f(x) = x^{9} + 3x^{7} + 64 is increasing on