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Question

If fx is differentiable and f''0=a, then limx02fx-3f2x+f4xx2=


A

3a

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B

2a

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C

5a

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D

4a

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Solution

The correct option is A

3a


Explanation for the correct option.

Find the value of the given limit.

The expression 2fx-3f2x+f4xx2 is of the form 00 at x0, so its limit can be found as:

limx02fx-3f2x+f4xx2=limx02f'x-3f'2x×2+f'4x×42x=limx02f'x-6f'2x+4f'4x2x=limx0f'x-3f'2x+2f'4xx

Again, the expression f'x-3f'2x+2f'4xxis of the form 00 at x0, so its limit can be found as:

limx0f'x-3f'2x+2f'4xx=limx0f''x-3f''2x×2+2f''4x×41=limx0f''(x)-6f''(x)+8f''(x)=limx03f''(x)=3f''(0)=3×a[f''(0)=a]=3a

Thus the value of limx02fx-3f2x+f4xx2 is 3a.

Hence, the correct option is A.


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