Find the absolute maximum value and the absolute minimum value

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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.


Let straight f left parenthesis straight x right parenthesis space equals space open parentheses 1 half minus straight x close parentheses squared space plus straight x space cubed

The given function is differentiable for all x in [-2, 2.5] 

 straight f apostrophe left parenthesis straight x right parenthesis space equals space 2 space open parentheses 1 half minus straight x close parentheses space left parenthesis negative 1 right parenthesis space plus space 3 straight x space equals space minus 1 plus 2 straight x plus 3 straight x squared
straight f apostrophe left parenthesis straight x right parenthesis space equals space 0 space space space space space space rightwards double arrow space space space space space space space 3 straight x squared plus 2 straight x minus 1 space equals space 0
rightwards double arrow space space space space space space straight x space equals space fraction numerator negative 2 space plus-or-minus square root of 4 plus 12 end root over denominator 6 end fraction space equals space fraction numerator negative 2 plus-or-minus 4 over denominator 6 end fraction equals negative 1 comma space 1 third
Both space space minus 1 comma space space space 1 third space space space element of space space space left square bracket negative 2 comma space 2.5 right square bracket

straight f left parenthesis negative 2 right parenthesis space equals space open parentheses 1 half plus 2 close parentheses squared plus left parenthesis negative 2 right parenthesis cubed space equals space 25 over 4 minus 8 space equals space minus 7 over 4 space equals space minus 1.75
straight f left parenthesis 2.5 right parenthesis space equals space open parentheses 1 half minus 2.5 close parentheses squared plus space left parenthesis 2.5 right parenthesis cubed space equals space 4 plus 125 over 8 space equals space 157 over 8 space equals space 19.625
straight f left parenthesis negative 1 right parenthesis space equals space open parentheses 1 half plus 1 close parentheses squared space minus 1 space equals space 9 over 4 minus 1 space equals space 5 over 4
straight f open parentheses 1 third close parentheses space equals space open parentheses 1 half minus 1 third close parentheses squared plus 1 over 27 space equals space 1 over 36 plus 1 over 27 space equals space fraction numerator 3 plus 4 over denominator 108 end fraction space equals space 7 over 108
therefore absolute maximum value = 19.625 and absolute minimum value  = -1.75

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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


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