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Application of Derivatives

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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

Let f(x) = 16x²– 16x + 28 = 16(x²- x) + 28
$equals 16 open parentheses straight x squared minus straight x plus 1 fourth close parentheses space plus space left parenthesis 28 minus 4 right parenthesis space equals space 16 open parentheses straight x minus 1 half close parentheses squared plus 24$
Now, $open parentheses straight x minus 1 half close parentheses squared space greater or equal than space 0 space space space for all space space space straight x space space space element of space space straight R space space space space space space rightwards double arrow space space space space space space 16 open parentheses straight x minus 1 half close parentheses squared space greater or equal than space 0 space space space for all space space space straight x space space element of space space straight R$
$rightwards double arrow space 16 space open parentheses straight x minus 1 half close parentheses squared plus 24 space greater or equal than space 24 space space for all space space straight x space space space element of space space straight R space space space rightwards double arrow space space space space straight f left parenthesis straight x right parenthesis space greater or equal than space 24 space for all space space straight x space space space element of space space straight R$
$therefore$ minimum value of f(x) is 24. It has no maximum value.

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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

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:
$straight f left parenthesis straight x right parenthesis space equals space 4 straight x minus 1 half straight x squared comma space space space straight x space element of space open square brackets negative 2 comma space space 9 over 2 close square brackets$