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Question

Verify Rolle's theorem for the function f(x) = x(x −2)2 on the interval [0, 2].


Solution

The given function is fx=xx-22, which can be rewritten as fx=x3-4x2+4x.

We know that a polynomial function is everywhere derivable and hence continuous.

So, being a polynomial function, fx is continuous and derivable on 0, 2

Also,
 f0=f2=0

Thus, all the conditions of Rolle's theorem are satisfied.

Now, we have to show that there exists c0, 2 such that f'c=0.

We have
fx=x3-4x2+4xf'x=3x2-8x+4When f'x=0  3x2-8x+4=03x2-6x-2x+4=03xx-2-2x-2=0x-23x-2x=2, 23

Thus, c=230, 2 such that f'c=0.

Hence, Rolle's theorem is verified.

Mathematics
RD Sharma XII Vol 1 (2015)
Standard XII

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