Q102 / 180 · KCET 2019 · Mathematics
Rolle's theorem is not applicable in which one of the following cases?
- A\( f(x)=x^{2}-x \) in \( [0,1] \)
- B\( f(x)=[x] \) in \( [2.5,2.7] \)
- C\( f(x)=x^{2}-4 x+5 \) in \( [1,3] \)
- Correct answerD\( f(x)=|x| \) in \( [-2,2] \)
Solution
\(f(x)=x^{2}-4 x+5\)
\(x \in[1,3]\)
\(\begin{aligned} f(1) &=1-4+5 \\ &=2 \\ f(3) &=9-12+5 \\ &=2 \\ f^{\prime}(x) &=2 x-4 \\ \lim _{h \rightarrow 0} f^{\prime}(x) &=\lim _{h \rightarrow 0}(2 x-4) \\ &=-4 \end{aligned}\)
\(\therefore\) IInd condition of rolls theorem is not satisfied.
\(\therefore\) Role theorem not applicable
Hence, the correct answer is option: 'D'.