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
$f(x) = x^2 - 8x + 12$ on $[2, 6]$

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Answer

Given: $f(x) = x^2 - 8x + 12$ We know that a polynomial function is everywhere derivable and hence continuous.
So, being a polynomial function f(x) is continuous and derivable on$ [2, 6].$
$f(2) = (2)^2 - 8(2) + 12 = 4 - 16 + 12 = 0$
$f(6) = (6)^2 - 8(6) + 12 = 36 - 48 + 12 = 0$
$\therefore$ $f(2) = f(6) = 0$
Thus, all the conditions of rolle's theorem are satisfied.
Now, we have to show that there exist $\text{c}\in(2, 6)$ such that f'(c) = 0
We have
$f(x) = x^2 - 8x + 12$
$\Rightarrow f'(x) = 2x - 8$
$\therefore$ $f'(x) = 0$
$\Rightarrow 2x - 8 = 0$
$\Rightarrow x = 4$
Thus, $\text{c}=4\in(2,6)$ such that f'(c) = 0

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