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Rolle's theorem is not applicable in which one of the following cases?
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The correct answer is:
\( f(x)=|x| \) in \( [-2,2] \)
\(f(x)=x^{2}-4 x+5\)
\(x \in[1,3]\)
\(\begin{aligned} f(1) &=1-4+5 \\ &=2 \\ f(3) &=9-12+5 \\ &=2 \\ f^{\prime}(x) &=2 x-4 \\ \lim _{h \rightarrow 0} f^{\prime}(x) &=\lim _{h \rightarrow 0}(2 x-4) \\ &=-4 \end{aligned}\)
\(\therefore\) IInd condition of rolls theorem is not satisfied.
\(\therefore\) Role theorem not applicable
\(x \in[1,3]\)
\(\begin{aligned} f(1) &=1-4+5 \\ &=2 \\ f(3) &=9-12+5 \\ &=2 \\ f^{\prime}(x) &=2 x-4 \\ \lim _{h \rightarrow 0} f^{\prime}(x) &=\lim _{h \rightarrow 0}(2 x-4) \\ &=-4 \end{aligned}\)
\(\therefore\) IInd condition of rolls theorem is not satisfied.
\(\therefore\) Role theorem not applicable
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