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Application of Derivatives

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Short Answer Type

151. Which of the following function are strictly decreasing on

open parentheses 0 comma space straight pi over 2 close parentheses space ?

(a) cos x (b) cos 2x (c) cos 3x (d) tan x

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152. Show that the function f (x) = x⁷ + 8 x⁵ + 1 is increasing function for all values of x.

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153.

Show that f (x) = tan ^-¹ (sin x + cos x) is a strictly increasing function in the interval $open parentheses 0 comma space space straight pi over 4 close parentheses$

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154.
Determine whether $straight g left parenthesis straight x right parenthesis space equals space cos space left parenthesis 2 straight x space plus space straight pi over 4 right parenthesis$ is increasing or decreasing for $fraction numerator 3 straight pi over denominator 8 end fraction less than straight x less than fraction numerator 7 straight pi over denominator 8 end fraction.$

Here $straight g left parenthesis straight x right parenthesis space equals space cos space open parentheses 2 straight x plus straight pi over 4 close parentheses$
$therefore space space space space straight g apostrophe left parenthesis straight x right parenthesis space equals space minus 2 space sin space open parentheses 2 straight x plus straight pi over 4 close parentheses For space space fraction numerator 3 straight pi over denominator 8 end fraction less than straight x less than fraction numerator 7 straight pi over denominator 8 end fraction comma space space space space straight g apostrophe left parenthesis straight x right parenthesis space is space positive$ .
$therefore space space space space straight g left parenthesis straight x right parenthesis space is space increasing space for space fraction numerator 3 straight pi over denominator 8 end fraction less than straight x less than fraction numerator 7 straight pi over denominator 8 end fraction.$

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155. Prove that the function log sin x is strictly increasing on

open parentheses 0 comma space space straight pi over 2 close parentheses

and strictly decreasing in

open parentheses straight pi over 2 comma space space straight pi close parentheses.

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156.

Prove that the function f given by f(x) = log cos x is strictly decreasing on $open parentheses 0 comma space straight pi over 2 close parentheses$ and strictly increasing on $open parentheses straight pi over 2 comma space straight pi close parentheses.$

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157. On which of the following intervals is the function x¹⁰⁰ + sin x – 1 strictly increasing?
(i) (1, 1) (ii) (0, 1) (iii)

open parentheses straight pi over 2 comma space straight pi close parentheses

(iv)

open parentheses 0 comma space space straight pi over 2 close parentheses

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158.

Find the value of a for which the function f (x) = x³ – 2ax + 6 is increasing for x > 0.

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159.

Show that $straight y equals log space left parenthesis 1 plus straight x right parenthesis space minus space fraction numerator 2 straight x over denominator 2 plus straight x end fraction$ is an increasing function of x throughout its domain.