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) = x² -4x + 3 on [1, 3]

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Answer

The given function is f(x) = x² -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) = 1² - 4 × 1 + 3 = 0
f(4) = 4² - 4 × 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
⇒ 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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