AP EAMCET · Maths · Application of Derivatives
If \(f(x)=x^{\mathrm{x}}\), then the interval in which \(f(x)\) decreases is
- A \(\left[0, \frac{1}{e}\right]\)
- B \([0, e]\)
- C \(\left[\frac{1}{e}, \infty\right]\)
- D \(\left[0, e^e\right]\)
Answer & Solution
Correct Answer
(A) \(\left[0, \frac{1}{e}\right]\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \text { } f(x)=x^x \Rightarrow y=x^x \\ & \log y=x \log x \\ & \frac{1}{y} \frac{d y}{d x}=\log x+1 \Rightarrow \frac{d y}{d x}=x^x(1+\log x) \end{aligned}\) Since, \(x^x\) is always +ve \(\therefore f(x)\) decreases when \(1+\log \leq 0\)…
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