Verify Rolle's theorem for function y=x^{2}+2, a=-2 and b=2

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5. Continuity and Differentiation

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Verify Rolle's theorem for the function $y=x^{2}+2, a=-2$ and $b=2$

Option A

Option B

Option C

Option D

Solution

The function $y=x^{2}+2$ is continuous in $[-2,2]$ and differentiable in $(-2,2).$

Also $f(-2)=f(2)=6$ and hence the value of $f(x)$ at $-2$ and $2$ coincide. Rolle's theorem states that there is a point $c \in(-2,2),$ where $f^{\prime}(c)=0 .$ Since $f^{\prime}(x)=2 x,$ we get $c=0 .$ Thus at $c=0,$ we have $f^{\prime}(c)=0$ and $c=0 \in(-2,2)$

Standard 12

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