Q110 / 160 · TS EAMCET 2024 (10 May Shift 2) · Mathematics

If Rolle's Theorem is applicable for the function $f(x)=\left\{\begin{array}{cc}x^p \log x, & x \neq 0 \\ 0, & x=0\end{array}\right.$ on the interval $[0,1]$, then a possible value of $p$ is

  • A
    -2
  • B
    -1
  • C
    0
  • Correct answer
    D
    1

Solution

Rolle's theorem is applicable on $f(x)$ $\begin{aligned} & \Rightarrow f(x) \text { is continuous on }[0,1] \\ & \Rightarrow \lim _{x \rightarrow 0} f(x)=0 \Rightarrow \lim _{x \rightarrow 0} x^p \log ^x=0 \\ & \Rightarrow \lim _{x \rightarrow 0} \frac{\log x}{x^{-p}}=0 \Rightarrow \lim _{x \rightarrow 0} \frac{\frac{1}{x}}{-p x^{-p-1}}=0 \\ & \Rightarrow \lim _{x \rightarrow 0} \frac{x^p}{-p}=0\end{aligned}$ Only possible when $p\gt0 \Rightarrow p=1$

Hence, the correct answer is option: 'D'.

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