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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

91 Views

222.

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

101 Views

223.

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

86 Views

224.

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

81 Views

225.

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

77 Views

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].

91 Views

Long Answer Type

227. Find absolute maximum and minimum values of a function f given by
$straight f left parenthesis straight x right parenthesis space equals space 12 straight x to the power of 4 over 3 end exponent space minus space 6 straight x to the power of 1 third end exponent comma space space space straight x space element of space space space open square brackets negative 1 comma space 1 close square brackets$

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 straight f apostrophe left parenthesis straight x right parenthesis space equals 12 space cross times 4 over 3 straight x to the power of 1 third end exponent space minus space 6 space cross times space 1 third straight x to the power of negative 2 over 3 end exponent space equals space 16 straight x to the power of 1 third end exponent minus 2 over straight x to the power of begin display style 2 over 3 end style end exponent space equals space fraction numerator 16 straight x minus 2 over denominator straight x to the power of begin display style 2 over 3 end style end exponent end fraction space equals space fraction numerator 2 left parenthesis 8 straight x minus 1 right parenthesis over denominator straight x to the power of begin display style 2 over 3 end style end exponent end fraction straight f apostrophe left parenthesis straight x right parenthesis space equals 0 space space space rightwards double arrow space fraction numerator 2 left parenthesis 8 straight x minus 1 right parenthesis over denominator straight x to the power of begin display style 2 over 3 end style end exponent end fraction space equals 0 space space space rightwards double arrow space space space space space 2 left parenthesis 8 straight x minus 1 right parenthesis space equals space 0 space space space space space rightwards double arrow space space space straight x space equals space 1 over 8 Also space straight f apostrophe left parenthesis straight x right parenthesis space is space not space defined space at space straight x space equals space 0

therefore space space space space space space critical space points space are space straight x space equals space 0 comma space space 1 over 8

Now,

Now comma space space space straight f left parenthesis negative 1 right parenthesis space equals space 12 left parenthesis negative 1 right parenthesis to the power of 4 over 3 end exponent space minus space 6 left parenthesis negative 1 right parenthesis to the power of 1 third end exponent space equals space 12 plus 6 space equals space 18 space space space space space space space space space space space space space straight f left parenthesis 0 right parenthesis space equals space 0 minus 0 space equals space 0 space space space straight f open parentheses 1 over 8 close parentheses space equals space 12 open parentheses 1 over 8 close parentheses to the power of 4 over 3 end exponent space minus space 6 open parentheses 1 over 8 close parentheses to the power of 1 third end exponent space equals space 12 left parenthesis 2 to the power of negative 3 end exponent right parenthesis to the power of 4 over 3 end exponent space minus space 6 left parenthesis 2 to the power of negative 3 end exponent right parenthesis to the power of 1 third end exponent space space space space space space space space space space space space space equals 12 left parenthesis 2 right parenthesis to the power of negative 4 end exponent space minus 6 space left parenthesis 2 right parenthesis to the power of negative 1 end exponent space equals space 12 space cross times space 1 over 16 minus 6 cross times 1 half equals space 3 over 4 minus 3 space equals space minus 9 over 4 space space space space space space space straight f left parenthesis 1 right parenthesis space equals space 12 left parenthesis 1 right parenthesis to the power of 4 over 3 end exponent space minus space 6 left parenthesis 1 right parenthesis to the power of 1 third end exponent space equals space 12 minus 6 space equals space 6

therefore space space space space absolute space maximum space value space space equals space 18 space space at space straight x space equals space minus 1 and space absolute space minimum space value space space equals space minus 9 over 4 space at space straight x space equals space 1 over 8.

73 Views

Short Answer Type

228.

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

87 Views

229.

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

77 Views

230.

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

86 Views