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Question

If the function f(x)=x3-6x2+ax+b satisfies Rolle's theorem in the interval 1,3 and f'23+13=0, then:


A

a=-11

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B

a=-6

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C

a=6

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D

a=11

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Solution

The correct option is D

a=11


Explanation for correct option:

Rolle's Theorem:

Given function, f(x)=x3-6x2+ax+b

According to Rolle's theorem, if a function f is continuous on the closed interval a,b and differentiable on the open interval a,b such that f(a)=f(b), then f'(x)=0 for some x with axb.

So, upon differentiating the given function, we get,

f'(x)=3x2-12x+a

Now, as it is given that the function satisfies the Rolle's theorem, so, f'(c)=0

f'23+13=0

We can replace this in f'(x)=3x2-12x+a,

0=32+132-122+13+a0=34+13+43-24-123+a0=12+1+43-24-43+a0=-11+aa=11

Therefore, option (D) is the correct answer.


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