Find the local maxima or local minima, if any, of following func

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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

233.

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

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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



Let straight f left parenthesis straight x right parenthesis space equals space sin space 2 straight x comma space space space 0 less than straight x less than straight pi
    therefore space space space straight f space apostrophe left parenthesis straight x right parenthesis space equals space 2 space cos space 2 straight x
space space space space space space space straight f space apostrophe left parenthesis straight x right parenthesis space equals space 0 space space space space space space space space space rightwards double arrow space space space 2 space cos space 2 straight x space equals space 0 space space space space space rightwards double arrow space space cos space 2 straight x space equals space 0
rightwards double arrow space space space space 2 straight x space equals space straight pi over 2 comma space space space fraction numerator 3 straight pi over denominator 2 end fraction space space space space rightwards double arrow space space space space space space straight x space equals space straight pi over 4 comma space space fraction numerator 3 straight pi over denominator 4 end fraction
space space space when space straight x space less than space straight pi over 4 space slightly space comma space space space straight f space apostrophe left parenthesis straight x right parenthesis space equals space 2 left parenthesis plus ve right parenthesis space equals space plus ve
When space straight x greater than straight pi over 4 space slightly comma space straight f space apostrophe left parenthesis straight x right parenthesis space equals 2 left parenthesis negative ve right parenthesis space equals space minus ve



space space space space space space

therefore space space space space space at space straight x space equals space straight pi over 4 comma space space space straight f apostrophe left parenthesis straight x right parenthesis space changes space from space plus ve space to space minus ve
therefore space space space space space space straight f left parenthesis straight x right parenthesis space has space local space maxima space at space straight x space equals space straight pi over 4
space and space local space maximum space value space space equals space sin space straight pi over 2 space equals space 1
space space space When space straight x space less than fraction numerator 3 straight pi over denominator 4 end fraction comma space space slightly space straight f apostrophe left parenthesis straight x right parenthesis space equals space 2 left parenthesis negative ve right parenthesis space equals space minus ve
space space When space straight x greater than space space fraction numerator 3 straight pi over denominator 4 end fraction space slightly comma space space straight f apostrophe left parenthesis straight x right parenthesis space equals space 2 left parenthesis plus ve right parenthesis space equals space plus ve
space space space When space straight x greater than fraction numerator 3 straight pi over denominator 4 end fraction comma space space straight f apostrophe left parenthesis straight x right parenthesis space changes space from space minus ve space to space ve
therefore space space space space straight f left parenthesis straight x right parenthesis space has space local space minima space at space straight x space equals space fraction numerator 3 straight pi over denominator 4 end fraction
space and space local space minimum space value space space equals space sin space fraction numerator 3 straight pi over denominator 2 end fraction space equals space minus sin space straight pi over 2 space equals space minus 1

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