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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

(a) Let $straight f left parenthesis straight x right parenthesis space equals space cos space straight x comma space space space space space space space space therefore space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus sin space straight x space$
In $open parentheses 0 comma space straight pi over 2 close parentheses comma space space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus sin space straight x space less than space 0$
$therefore space space space space straight f left parenthesis straight x right parenthesis space equals space cosx space is space strictly space decreasing space on space open parentheses 0 comma space straight pi over 2 close parentheses$

(b) Let $straight f left parenthesis straight x right parenthesis space equals space cos space 2 straight x comma space space space space space space therefore space space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus 2 space sin space 2 straight x$
In $open parentheses 0 comma space straight pi over 2 close parentheses comma space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus 2 space sin space 2 straight x space less than space 0$
$therefore space space space space space straight f left parenthesis straight x right parenthesis space equals space cos space 2 straight x$ is strictly decreasing on $open parentheses 0 comma space straight pi over 2 close parentheses.$
(c) Let f(x) = cos 3x, $therefore space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus 3 space sin space 3 straight x$
$0 less than straight x less than straight pi over 2$
$rightwards double arrow space space space space space space 0 less than space 3 straight x space less than space fraction numerator 3 straight pi over denominator 2 end fraction$
Now $straight f apostrophe left parenthesis straight x right parenthesis space less than 0 space space space space when space 0 space less than space 3 straight x space less than space straight pi$
and $straight f apostrophe left parenthesis straight x right parenthesis space greater than 0 space space space when space straight pi space less than space 3 straight x space less than fraction numerator 3 straight pi over denominator 2 end fraction$
$therefore space space space space space straight f left parenthesis straight x right parenthesis space is space not space straight a space strictly space decreasing space function space on space open parentheses 0 comma space space straight pi over 2 close parentheses$

(d) Let $straight f left parenthesis straight x right parenthesis space equals tanx comma space space space space space space space space therefore space space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space sec squared straight x$
Now, $straight f apostrophe left parenthesis straight x right parenthesis greater than 0 space space for all space straight x space space space space element of space space space open parentheses 0 comma space space straight pi over 2 close parentheses$
$therefore space space space straight f left parenthesis straight x right parenthesis space is space not space straight a space strictly space decreasing space function space on space open parentheses 0 comma space space space straight pi over 2 close parentheses$

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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 $fraction numerator 3 straight pi over denominator 8 end fraction less than straight x less than fraction numerator 7 straight pi over denominator 8 end fraction.$

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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.