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Application of Derivatives

Name: Zigya - For The Curious Learner
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Short Answer Type

221.

Find the maximum or minimum values, if any, of the following functions without using the derivatives:
16x²– 16x + 28

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222.

Prove that the following functions do not have maxima or minima:
f(x) = e^x

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223.

Prove that the following functions do not have maxima or minima:
g(x) = log x

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224.

Prove that the following functions do not have maxima or minima:
h(x) = x³ + x² + x + 1

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225.
Find the absolute maximum value and the absolute minimum value of
$straight f left parenthesis straight x right parenthesis space equals space 1 third straight x cubed minus 3 straight x squared plus 5 straight x plus 8 space in space left square bracket 0 comma space 4 right square bracket.$

The function $straight f left parenthesis straight x right parenthesis space equals space 1 third straight x cubed minus 3 straight x squared plus 5 straight x plus 8$ is differentiable for all x in $open square brackets 0 comma space 4 close square brackets$
and $straight f apostrophe left parenthesis straight x right parenthesis space equals space straight x squared minus 6 straight x plus 5$
Now, $straight f apostrophe left parenthesis straight x right parenthesis space equals space 0 space space space space space space when space straight x squared minus 6 straight x plus 5 space equals space 0$
i.e., when (x - 5) (x - 1) = 0
i.e., when x = 1 or x = 5
But only $1 space element of space space space open square brackets 0 comma space 4 close square brackets$
Now, $straight f left parenthesis 1 right parenthesis space equals space 1 third left parenthesis 1 right parenthesis cubed space minus space 3. space space left parenthesis 1 right parenthesis squared space plus space 5 space left parenthesis 1 right parenthesis space plus space 8 space equals space 31 over 3$
$straight f left parenthesis 0 right parenthesis space equals space 8 comma space space space space space straight f left parenthesis 4 right parenthesis space equals space 1 third left parenthesis 4 right parenthesis cubed space minus space 3 space left parenthesis 4 right parenthesis squared space plus space 5 space left parenthesis 4 right parenthesis space plus space 8 space equals space 4 over 3.$
Hence absolute maximum value is $31 over 3$ and absolute minimum value is $4 over 3$ . These are attained at 1 and 4 respectively.

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226.

Find the absolute maximum and minimum values of a function f is given by
f (x) = 2x³ – 15x² + 36x + 1 on the interval of [1, 5].

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Long Answer Type

227. Find absolute maximum and minimum values of a function f given by

straight f left parenthesis straight x right parenthesis space equals space 12 straight x to the power of 4 over 3 end exponent space minus space 6 straight x to the power of 1 third end exponent comma space space space straight x space element of space space space open square brackets negative 1 comma space 1 close square brackets

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Short Answer Type

228.

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
$straight f left parenthesis straight x right parenthesis space equals space straight x cubed comma space space space straight x space element of space left square bracket negative 2 comma space space 2 right square bracket$

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229.

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
$straight f left parenthesis straight x right parenthesis space equals space sin space straight x plus space cosx comma space space straight x space space element of space space left square bracket 0 comma space straight pi right square bracket$

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230.

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
$straight f left parenthesis straight x right parenthesis space equals space 4 straight x minus 1 half straight x squared comma space space space straight x space element of space open square brackets negative 2 comma space space 9 over 2 close square brackets$