Find the absolute maximum and minimum values of the function f g

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 Multiple Choice QuestionsShort Answer Type

231.

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
straight f left parenthesis straight x right parenthesis space equals space left parenthesis straight x minus 1 right parenthesis squared plus 3 comma space space space space straight x space element of space space open square brackets negative 3 comma space space 1 close square brackets



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232.

Find the absolute maximum value and the absolute minimum value of
straight f left parenthesis straight x right parenthesis space equals space open parentheses 1 half minus straight x close parentheses squared space space plus space straight x space cubed space space space in space space space left square bracket negative 2 comma space space 2.5 right square bracket.

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 Multiple Choice QuestionsLong Answer Type

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233.

Find the absolute maximum and minimum values of the function f given by
 f (x) = cos2x + sinx,  x ∊ [0, straight pi].


straight f left parenthesis straight x right parenthesis space equals cos squared straight x plus space sinx
It is differentiable for all x in open square brackets 0 comma space straight pi close square brackets.
straight f apostrophe left parenthesis straight x right parenthesis space equals space 2 space cos space straight x left parenthesis negative sin space straight x right parenthesis space space plus space space cos space straight x space space equals space cosx space left parenthesis 1 minus 2 space sin space straight x right parenthesis
straight f apostrophe left parenthesis straight x right parenthesis space equals space 0 space gives space cosx left parenthesis 1 minus 2 space sinx right parenthesis space equals space 0
therefore space space space space space cosx space equals space 0
therefore space space space space space straight x space equals space straight pi over 2

or space space space 1 minus 2 space sinx space space equals space 0
therefore space space space space space sin space straight x space equals space 1 half
therefore space space space sin space straight x space equals space sin space straight pi over 6 comma space space sin space fraction numerator 5 straight pi over denominator 6 end fraction
therefore space space space space space space space straight x space equals space straight pi over 6 comma space space fraction numerator 5 straight pi over denominator 6 end fraction
therefore space space space space space space space straight x space equals space straight pi over 3 comma space straight pi over 2 comma space fraction numerator 5 space straight pi over denominator 6 end fraction
  At straight x equals straight pi over 6 comma space space straight f left parenthesis straight x right parenthesis space equals space cos squared straight pi over 6 plus sin straight pi over 6 space equals open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses squared plus 1 half space equals space 3 over 4 plus 1 half space equals space 5 over 4
 
At space space straight x space equals space fraction numerator 5 straight pi over denominator 6 end fraction comma space space space space straight f left parenthesis straight x right parenthesis space equals space cos squared fraction numerator 5 straight pi over denominator 6 end fraction plus sin space fraction numerator 5 straight pi over denominator 6 end fraction space equals space cos squared open parentheses straight pi minus straight pi over 6 close parentheses plus sin space open parentheses straight pi minus straight pi over 6 close parentheses
space space space space space space space space space space space equals space cos squared straight pi over 6 plus sin space straight pi over 6 space equals space 5 over 4
space At space straight x space equals space straight pi over 2 comma space space space straight f left parenthesis straight x right parenthesis space equals space cos squared straight pi over 2 plus sin space straight pi over 2 space equals space 0 plus 1 space equals space 1
At space space space straight x space equals space 0 comma space space space straight f left parenthesis straight x right parenthesis space equals space cos squared straight pi over 4 space plus space sin space straight pi space equals space left parenthesis negative 1 right parenthesis squared plus 0 space equals space 1 plus 0 space equals space 1
therefore space absolute space max space space value space space equals space 5 over 4 space and space aboslute space minimum space value space space equals space 1
 
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234.

Find the points at which the function f given by f (x) = (x – 2)4 (x + 1 )3 has
(i) local maxima (ii) local minima (iii) point of inflexion .

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235.

Examine the following function for extreme values:
f(x) = (x – 3)(x + 1)6

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 Multiple Choice QuestionsShort Answer Type

236.

Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
The constant function straight alpha

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237.

Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
f(x) = x2

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238.

Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
straight x cubed minus 3 space straight x

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239.

Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
cosx comma space space 0 space less than space straight x space less than straight pi

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240.

Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
sin space 2 straight x comma space space 0 space less than space straight x space less than straight pi


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