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
$f(x) = x^2 -4x + 3$ on $[1, 3]$

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Answer

The given function is $f(x) = x^2 -4x + 3$
f, being a pollynomial function, is continuous in [1, 4] and is differentiable in (1, 4) whose derivative is $2x - 4.$
$f(1) = 1^2 - 4 \times 1 + 3 = 0$
$f(4) = 4^2 - 4 \times 4 + 3 = 3$
$\therefore\ \frac{\text{f}(\text{b})-\text{f}(\text{a})}{\text{b}-\text{a}}=\frac{\text{f}(4)-\text{f}(1)}{4-1}=\frac{3-(0)}{3}=\frac{3}{3}=1$
Mean Value Theorem states that there is a point $\text{c}\in(1,4)$ such that $f'(c) = 1$
$f'(c) = 1$
$\Rightarrow 2c - 4 = 1$
$\Rightarrow\text{c}=\frac{5}{2},$ where $\text{c}=\frac{5}{2}\in(1,4)$
Hence, Mean Value Theorem is verified for the given function.

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