Show that  is an increasing function of x throughout its domai

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 Multiple Choice QuestionsShort 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.

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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.


straight y space equals space 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
therefore space space space space space dy over dx space equals space fraction numerator 1 over denominator 1 plus straight x end fraction minus fraction numerator left parenthesis 2 plus straight x right parenthesis. space 2 space minus space 2 space straight x. space 1 over denominator left parenthesis 2 plus straight x right parenthesis squared end fraction space equals space fraction numerator 1 over denominator 1 plus straight x end fraction space equals space fraction numerator 4 plus 2 straight x minus 2 straight x over denominator left parenthesis 2 plus straight x right parenthesis squared end fraction space equals fraction numerator 1 over denominator 1 plus straight x end fraction minus fraction numerator 4 over denominator left parenthesis 2 plus straight x right parenthesis squared end fraction
space space space space space space space space space space space space space space space space space space space space equals fraction numerator left parenthesis 2 plus straight x right parenthesis squared minus 4 left parenthesis 1 plus straight x right parenthesis over denominator left parenthesis 1 plus straight x right parenthesis thin space left parenthesis 2 plus straight x right parenthesis squared end fraction space equals space fraction numerator 4 plus straight x squared plus 4 straight x minus 4 minus 4 straight x over denominator left parenthesis 1 plus straight x right parenthesis thin space left parenthesis 2 plus straight x right parenthesis squared end fraction space equals space fraction numerator straight x squared over denominator left parenthesis 1 plus straight x right parenthesis thin space left parenthesis 2 plus straight x right parenthesis squared end fraction greater than 0
                                                                                                          open square brackets because space space space space straight x space equals space minus 1 close square brackets
therefore space space space space straight y space equals space 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 space is space an space increasing space function space of space straight x space for all space straight x space space equals space minus 1.
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160.

Find the values of x for which y = [x ( x – 2)]2 is an increasing function.

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