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Application of Derivatives

Name: Zigya - For The Curious Learner
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Multiple Choice Questions

201.
The interval in which y = x²e^–x is increasing is

(– ∞, ∞)
(– 2, 0)
(2, ∞)
(2, ∞)

(2, ∞)

Here, $straight y space equals space straight x squared straight e to the power of negative straight x end exponent$
$therefore space space space space space dy over dx space equals space straight x squared straight d over dx left parenthesis straight e to the power of negative straight x end exponent right parenthesis space plus space straight e to the power of negative straight x end exponent straight d over dx left parenthesis straight x squared right parenthesis space equals space straight x squared. space straight e to the power of negative straight x end exponent straight d over dx left parenthesis negative straight x right parenthesis space plus straight e to the power of negative straight x end exponent. space 2 straight x$
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For y to be increasing,
$dy over dx greater than 0 space space space space space space space space space space space rightwards double arrow space space space minus straight x left parenthesis straight x minus 2 right parenthesis thin space straight e to the power of negative straight x end exponent greater than 0 space space space space rightwards double arrow space space space straight x left parenthesis straight x minus 2 right parenthesis space less than space 0 space space space rightwards double arrow space space space straight x space element of space space left parenthesis 0 comma space 2 right parenthesis$
$therefore space space space left parenthesis straight D right parenthesis space is space correct space answer.$

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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).

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205.

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

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206.

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

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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)

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209.

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

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210.

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

78 Views