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- 4)² on the interval [0, 4]

✓

Answer

Given function is f(x) = x(x- 4)². Which can be rewritten as f(x) = x³ - 8x² + 16x.
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, 4].
Also,
f(0) = f(4) = 0
Thus, all the conditions of Rolle's theorem are satisfied.
Now, we have to show that there exists $\text{c}\in[0,4]$ such that f'(c) = 0.
We have
f(x) = x³- 8x² + 16x
⇒ f'(x) = 3x²- 16x + 16
$\therefore$ f'(x) = 0
⇒ 3x²- 16x + 16 = 0
⇒ 3x²- 12x - 4x + 16 = 0
⇒ 3x(x - 4) - 4(x - 4) = 0
⇒ (x - 4)(3x - 4)
$\Rightarrow\text{x}=4,\frac{4}{3}$
Thus, $\text{c}=\frac{4}{3}\in(0,4)$ such that f'(c) = 0.
Hence, Rolle's theorem is verified.

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