KCET · Maths · Application of Derivatives
The maximum value of \( \left(\frac{1}{x}\right)^{x} \) is
- A \( \mathrm{e} \)
- B \( e^{e} \)
- C \( e^{1 / e} \)
- D \( \left(\frac{1}{e}\right)^{1 / e} \)
Answer & Solution
Correct Answer
(C) \( e^{1 / e} \)
Step-by-step Solution
Detailed explanation
Let us suppose \(f=x^{-x}\)
\(\Rightarrow \log f=-x \log x\)
\(\Rightarrow \frac{1}{f} \frac{d f}{d x}=-(1+\log x)\)
So, \(\frac{d f}{d x}=-f\left(1+\log _{e} x\right)\)
For maximum; \(1+\log _{e} x=0\)
\(\Rightarrow x=e^{-1}\)
\(\Rightarrow x=\frac{1}{e}\)
Therefore, maximum value of given function is
\(\left(\frac{1}{1 / e}\right)=e^{\frac{1}{e}}\)
\(\Rightarrow \log f=-x \log x\)
\(\Rightarrow \frac{1}{f} \frac{d f}{d x}=-(1+\log x)\)
So, \(\frac{d f}{d x}=-f\left(1+\log _{e} x\right)\)
For maximum; \(1+\log _{e} x=0\)
\(\Rightarrow x=e^{-1}\)
\(\Rightarrow x=\frac{1}{e}\)
Therefore, maximum value of given function is
\(\left(\frac{1}{1 / e}\right)=e^{\frac{1}{e}}\)
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
- \(\int \frac{\mathrm{dx}}{\mathrm{x}^2\left(\mathrm{x}^4+1\right)^{3 / 4}}\) equalsKCET 2025 Medium
- If \(A M\) and \(G M\) of roots of a quadratic equation are 5 and 4 , respectively, then the quadratic equation isKCET 2024 Easy
- The value of \(\sin \frac{5 \pi}{12} \sin \frac{\pi}{12}\) isKCET 2022 Easy
- The perimeter of a certain sector of a circle is equal to the length of the arc of the semicircle. Then, the angle at the centre of the sector in radians isKCET 2008 Easy
- Which of the following is incorrect?
If \(\mathrm{a} \equiv \mathrm{b}(\bmod \mathrm{m})\) and \(\mathrm{x}\) is an integer, thenKCET 2012 Easy - Suppose that the number of elements in set \(A\) is \(p\), the number of elements in set \(B\) is \(q\) and the number of elements in \(A \times B\) is 7 , then \(p^2+q^2=\)KCET 2022 Easy
More PYQs from KCET
- The time required for \(60 \%\) completion of a first order reaction is \(50 \mathrm{~min}\).
The time required for \(93.6 \%\) completion of the same reaction will beKCET 2020 Medium - The time for optimum chances of conception in a woman is __________ starting from the day menstruation.KCET 2012 Medium
- The simplified form of \( \tan ^{-1}\left(\frac{x}{y}\right)-\tan ^{-1}\left(\frac{x-y}{x+y}\right) \) is equal toKCET 2016 Medium
- The differential coefficient of \( \log _{10} x \) with respect to \( \log _{x} 10 \) isKCET 2016 Easy
- Dimensional formula for activity of radioactive substance isKCET 2024 Easy
- Seven identical discs are arranged in a planar pattern, so as to touch each other as shown in the figure. Each disc has mass \(m\) radius \(R\). What is the moment of inertia of system of six discs about an axis passing through the centre of central disc and normal to plane of all discs?
KCET 2023 Easy