Verify Rolle's theorem for the following function on the indicate… - Vidyadip

Question

Verify Rolle's theorem for the following function on the indicated intervals $f(x) = x(x - 2)^2$ on the interval $[0, 2]$

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Answer

Given function is $f(x) = x(x^- 2)^2.$ Which can be rewritten as $f(x) = x^3 - 4x^2 + 4x.$
We know that a polynomial function is everywhere derivable and hence continuous.
So, being a polynomial function $f(x) $ is continuous and derivable on $[0, 2].$
Also,
$f(0) = f(2) = 0$
Thus, all the conditions of Rolle's theorem are satisfied.
Now, we have to show that there exists $\text{c}\in[0,2]$ such that $ f'(c) = 0.$
We have
$f(x) = x^3- 4x^2 + 4x$
$\Rightarrow f'(x) = 3x^2- 8x + 4$
When, $f'(x) = 0$
$3x^2- 8x + 4 = 0$
$\Rightarrow 3x^2- 6x - 2x + 4 = 0$
$\Rightarrow 3x(x - 2) - 2(x - 2) = 0$
$\Rightarrow (x - 2)(3x - 2)$
$\Rightarrow\text{x}=2,\frac{2}{3}$
Thus, $\text{c}=\frac{2}{3}\in(0,2)$ such that $f'(c) = 0.$$$
Hence, Rolle's theorem is verified.

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