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Application of Derivatives

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].

f (x) = 2x³ – 15x² + 36x + 1
∴ f ' (x) = 6x²– 30x + 36 = 6 (x²– 5x + 6) = 6 (x – 2) (x – 3)
f ' (x) = 0 ⇒ 6 (x – 2) (x – 3) = 0 ⇒ x = 2, 3 ∊ [1,5]
Now f (1) = 2 – 15 + 36 + 1 = 24
f (2) = 2 × 8 – 15 x 4 + 36 x 2 + 1 = 16 – 60 + 72 + 1 = 29
f (3) = 2 × 27 – 15 x 9 + 36x3 + 1 = 54 – 135 + 108 + 1 = 28
f (5) = 2 × 125–15 × 25 + 36 x 5 + 1 = 250–375+ 180+ 1 = 56
∴ absolute maximum value = 56 at x = 5
and absolute minimum value = 24 at x = 1.

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