Verify Rolle’s theorem for the function f(x)=x^2+2x-8,x [-4,2]. - Vidyadip

Question

Verify Rolle’s theorem for the function $\text{f(x)}=\text{x}^2+2\text{x}-8,\text{x}\in[-4,2].$

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Answer

Consider $\text{f(x)}=\text{x}^2+2\text{x}-8,\text{x}\in[-4,\ 2]$
Function is continuous in [-4,2] as it is a polynomial function and polynomial function is always continuous.
$\text{f}'\text{(x)}=2\text{x}+2, \text{f}'\text{(x)}$ exists in [-4, 2], hence derivable.
$\text{f}(-4)=0\text{ and f }(2)=0$
$\therefore\ \text{f}(-4)=\text{f}(2)$
Conditions of Rolle’s theorem are satisfied, hence there exists, at least one $\text{c}\in(-4,2)$ such that $\text{f}'\text{(c)}=0$
$\Rightarrow\ 2\text{c}+2=0\ \Rightarrow\ \text{c}=-1$

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