Verify Mean Value Theorem, if f(x) = x^2 - 4x - 3 in the interval… - Vidyadip

Question

Verify Mean Value Theorem, if $f(x) = x^2 - 4x - 3$ in the interval $[a, b],$ where $a = 1$ and $b = 4$.

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Answer

Function is continuous in [1, 4] as it is a polynomial function and polynomial function is always continuous. f'(x) = 2x - 4, f'(x) exists in [1, 4], hence derivable. Conditions of MVT theorem are satisfied, hence there exists, at least one $\text{c}\in(1,\ 4)$ such that.
$\frac{\text{f}(4)-\text{(f)}(1)}{4-1}=\text{f}'\text{(c)}\ \Rightarrow\ \frac{-3-(-6)}{3}=2\text{c}-4$
$\Rightarrow\ 1=2\text{c}-4\ \Rightarrow\ \text{c}=\frac{5}{2}$

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