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

Question

Verify Rolle's theorem for the following function on the indicated intervals
$f(x) = x(x - 1)^2$^ on $[0, 1]$

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Answer

Given $f(x) = x(x - 1)^2$
$\Rightarrow f(x) = x(x^2 - 2x + 1)$
$\therefore f(x) = (x^3- 2x^2+ x)$
We know that a polynominal function is everywhere derivable and hence continuous.
So, being a polynomial function, f(x) is continuous and derivable on [0, 1]
Also,
f(0) = f(1) = 0
Thus, all the continuous of Solids theorem are satisfied.
Now, we have to show that there exists $\text{c}\in(0,1)$ such that f'(c) = 0.
We have
$f(x) = x^3 -^2x^2 +^x$
$\Rightarrow f'(x) = 3x^2 - 4x + 1$
$\therefore f'(x) = 0 \Rightarrow 3x^2- 4x + 1 = 0$
$\Rightarrow 3x^2- 3x - x + 1 = 0$
$\Rightarrow 3x(x - 1) - 1(x - 1) = 0$
$\Rightarrow (x - 1)(3x - 1) = 0$
$\Rightarrow\text{x}=1,\frac{1}{3}$
Thus, $\text{c}=\frac{1}{3}\in(0,1)$ such that f'(c) = 0.
Hence, Rolle's theorem is verified.

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