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Mean Value Theorems — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMean Value Theorems3 Marks
Question
Discuss the applicability of the Rolle's theorem for the following function on the indicated interval
$\text{f}(\text{x})=\begin{cases}-4\text{x}+5,&0\leq\text{x}\leq1\\2\text{x}-3,&1<\text{x}\leq2\end{cases}$
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Answer

The given function is
$\text{f}(\text{x})=\begin{cases}-4\text{x}+5,&0\leq\text{x}\leq1\\2\text{x}-3,&1<\text{x}\leq2\end{cases}$
At x = 0, we have
$\lim\limits_{\text{x}\rightarrow1^-}\text{f}(\text{x})=\lim\limits_{\text{h}\rightarrow0}\text{f}(1-\text{h})=\lim\limits_{\text{h}\rightarrow0}[-4(1-\text{h}+5)]=1$
$\lim\limits_{\text{x}\rightarrow1^+}\text{f}(\text{x})=\lim\limits_{\text{h}\rightarrow0}\text{f}(1+\text{h})=\lim\limits_{\text{h}\rightarrow0}[2(1+\text{h}-3)]=-1$
$\therefore\ \lim\limits_{\text{x}\rightarrow1^-}\text{f}(\text{x})\neq\lim\limits_{\text{x}\rightarrow1^+}\text{f}(\text{x})$
Thus, f(x) is discontinuous at x = 1.
Hence, Rolle's theorem is not applicable for the given function.

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