KCET · Maths · Application of Derivatives
\( f(x)=x^{x} \) has stationary point at
- A \( x=\theta \)
- B \( x=\frac{1}{e} \)
- C \( x=1 \)
- D \( \mathrm{x}=\sqrt{\mathrm{e}} \)
Answer & Solution
Correct Answer
(B) \( x=\frac{1}{e} \)
Step-by-step Solution
Detailed explanation
Given that, \(f(x)=x^{x}\)
\(\Rightarrow \log f(x)=x \log x\)
\(\Rightarrow \frac{1}{f(x)} f^{\prime}(x)=\log x+1\)
\(\Rightarrow f^{\prime}(x)=f(x)(\log x+1)=x^{x}(\log x+1)\)
Stationary point is given by
\(f^{\prime}(x)=0\)
\(x^{x}(\log x+1)=0\)
\(\Rightarrow \log x=-1\)
\(\Rightarrow x=e^{-1} \Rightarrow x=\frac{1}{e}\)
\(\Rightarrow \log f(x)=x \log x\)
\(\Rightarrow \frac{1}{f(x)} f^{\prime}(x)=\log x+1\)
\(\Rightarrow f^{\prime}(x)=f(x)(\log x+1)=x^{x}(\log x+1)\)
Stationary point is given by
\(f^{\prime}(x)=0\)
\(x^{x}(\log x+1)=0\)
\(\Rightarrow \log x=-1\)
\(\Rightarrow x=e^{-1} \Rightarrow x=\frac{1}{e}\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- \(\lim _{x \rightarrow 1} \frac{\tan \left(x^{2}-1\right)}{x-1}\) is equal toKCET 2007 Easy
- If \( \vec{a}, \vec{b}, \vec{c} \) are unit vectors such that \( \vec{a}+\vec{b}+\vec{c}=\overrightarrow{0} \) then the value of \( \vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a} \)
is equal toKCET 2017 Hard - A line passes through \( (2,2) \) and is perpendicular in the line \( 3 x+y=3 \) its \( y \)-intercepts isKCET 2015 Hard
- If \( { }^{n} C_{12}={ }^{n} C_{8} \), then \( \mathrm{n} \) is equal toKCET 2017 Hard
- Let \( f(x)=x-\frac{1}{x} \) then \( f^{\prime}(-1) \) isKCET 2018 Easy
- If a matrix \(A=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]\) satisfies \(A^6=k A^{\prime}\), then the value of \(k\) isKCET 2025 Medium
More PYQs from KCET
- Among the following, which is least acidic?KCET 2008 Easy
- A resistor has a colour code of green, blue, brown and silver. What is its resistance?KCET 2011 Easy
- In a face centred cubic arrangement of \( A \) and \( B \) atoms in which 'A' atoms are at the corners of
the unit cell and 'B' atoms are at the face centers. One of the 'A' atom is missing from one
corner in unit cell.the simplest formula of the compound isKCET 2017 Easy - If \( P(n): " 2^{2 n-1} \) is divisible by \( k \) for all \( n \varepsilon N^{\prime \prime} \) is true, then the value of ' \( k \) ' isKCET 2018 Medium
- If \( \left[\begin{array}{cc}1 & 1 \\ -1 & 1\end{array}\right]\left[\begin{array}{l}\mathrm{x} \\ \mathrm{y}\end{array}\right]=\left[\begin{array}{c}2 \\ 4\end{array}\right] \), then the values of \( \mathrm{x} \) and \( \mathrm{y} \) respectively areKCET 2018 Easy
- Two thin lenses have a combined power of +9D. When they are separated by a distance of \(20 \mathrm{~cm}\), their equivalent power becomes \(+\frac{27}{5}\) D. Their individual powers (in dioptre) areKCET 2011 Medium