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Differential Calculus

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Question : 23 of 69

Marks: +1, -0

If

f(x)=3 e^{-3 x}+2 e^{3 x}

f''(x)

is

$6 f(x)$
$3 f(x)$
$9 f(x)$
none

Solution:

Concept:

Differentiate twice and express the result in terms of

f(x)

.

Explanation:

First derivative:

f'(x) = 3 \cdot (-3e^{-3x}) + 2 \cdot (3e^{3x}) = -9e^{-3x} + 6e^{3x}

.

Second derivative:

f''(x) = -9 \cdot (-3e^{-3x}) + 6 \cdot (3e^{3x}) = 27e^{-3x} + 18e^{3x}

.

Factor out

9

:

f''(x) = 9(3e^{-3x} + 2e^{3x}) = 9f(x)

.

Answer:

9 f(x)

, which corresponds to option C.

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