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Application of Derivatives

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Multiple Choice Questions

201.

The interval in which y = x²e^–x is increasing is

(– ∞, ∞)
(– 2, 0)
(2, ∞)
(2, ∞)

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Short Answer Type

202.

Find the maximum and the minimum values, if there be any, of f given by f(x) = 9x²– 6x + 1, x ∊ R.

89 Views

203. Find the maximum and the minimum values of f, if there be any, given by
f(x) = – x², x ∊ R

87 Views

204.

Find the maximum and the minimum values, if there be any, of f given by f(x) = x, x ∊ (0, 1).

83 Views

205.
Without using derivatives, find the maximum or minimum value of the function 9x²+ 12x + 2.

$Let space straight f left parenthesis straight x right parenthesis space equals space 9 straight x squared plus 12 straight x plus 2 space equals space left parenthesis 9 straight x squared plus 12 straight x right parenthesis space plus 2$
$equals space 9 open parentheses straight x squared plus 4 over 3 straight x close parentheses space plus 2 space equals space 9 space open parentheses straight x squared plus 4 over 3 straight x plus 4 over 9 close parentheses plus left parenthesis 2 minus 4 right parenthesis space equals space 9 open parentheses straight x plus 2 over 3 close parentheses squared minus 2$

Now,

open parentheses straight x plus 2 over 3 close parentheses squared space greater or equal than space space 0 space space space for all space straight x space space element of space space straight R space space space space space rightwards double arrow space space space space 9 open parentheses straight x plus 2 over 3 close parentheses squared space greater or equal than space 0 space for all space straight x space space element of space space straight R

therefore space space space space space 9 open parentheses straight x plus 2 over 3 close parentheses squared minus 2 space greater or equal than space minus 2 space for all space space straight x space space element of space space straight R therefore space space space space space straight f left parenthesis straight x right parenthesis space greater or equal than negative 2 space space space space for all space space straight x space space element of space space straight R therefore space space minimum space value space of space straight f left parenthesis straight x right parenthesis space is space minus 2 space and space is space obtained space at space straight x space equals space minus 2 over 3. There space is space no space maximum space value space of space the space function. space

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206.

Find the maximum and minimum values, if any, of the following function without using the derivatives:
9x² – 12x + 4

97 Views

207.

Find the maximum and minimum values, if any, of the following function without using the derivatives:
4x ²+ 28x + 49

85 Views

208.

Find the maximum and minimum values, if any, of the following function without using the derivatives:
x + 1, x ∊ (– 1, 1)

92 Views

209.

Find the maximum and minimum values, if any, of the following functions without using the derivatives:
x²

89 Views

210.

Find the maximum and minimum values, if any, of the following functions without using the derivatives:
– (x – 2 )² + 4

78 Views