MCQ
The function $f(x) = {x^{1/x}}$ is
- AIncreasing in $(1,\,\,\infty )$
- BDecreasing in $(1,\,\,\infty )$
- ✓Increasing in $(1,\,e),$ decreasing in $(e,\infty )$
- DDecreasing in $(1,\,e),$ increasing in$(e,\infty )$
✓
Answer
Correct option: C.
Increasing in $(1,\,e),$ decreasing in $(e,\infty )$
c
(c) Let $y = {x^{1/x}}$==> $\log y = \frac{1}{x}\log x$
(c) Let $y = {x^{1/x}}$==> $\log y = \frac{1}{x}\log x$
==>$\frac{1}{y}\frac{{dy}}{{dx}} = \frac{1}{{{x^2}}} - \frac{{\log x}}{{{x^2}}} = \frac{{1 - \log x}}{{{x^2}}}$
==>$\frac{{dy}}{{dx}} = {x^{1/x}}\left( {\frac{{1 - \log x}}{{{x^2}}}} \right)$
Now, ${x^{1/x}} > 0$for all x and $\frac{{1 - \log x}}{{{x^2}}} > 0$ in $ (1, e) $ and
$\frac{{1 - \log x}}{{{x^2}}} < 0$ in $(e,\infty )$
$\therefore$ $f(x)$ is increasing in $ (1, e) $ and decreasing in $(e,\,\infty ).$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
The product of all the rational roots of the equation $\left(x^{2}-9 x+11\right)^{2}-(x-4)(x-5)=3$, is equal to :
→The sum $\frac{{3 \times 1}}{{{1^2}}} + \frac{{5 \times ({1^3} + {2^3})}}{{{1^2} + {2^2}}} + \frac{{7 \times ({1^3} + {2^3} + {3^3})}}{{{1^2} + {2^2} + {3^2}}} + .......$ Upto $10^{th}$ term, is
→If $z$ is a complex number such that $|z - \bar{z}| = 2$ and $|z + \bar{z}| = 4 $, then which of the following is always incorrect -
→The inequality to represent the coloured region in figure is....
$\sum\limits_{m = 1}^n {{m^2}} $ is equal to
→Solve $\sin ^{-1}(1-x)-2 \sin ^{-1} x=\frac{\pi}{2},$ then $x$ is equal to
→If $2\sec 2\alpha = \tan \beta + \cot \beta ,$ then one of the values of $\alpha + \beta $ is
→Let $A=\left[a_{i j}\right], a_{i j} \in Z \cap[0,4], 1 \leq i, j \leq 2$. The number of matrices $A$ such that the sum of all entries is a prime number $p \in(2,13)$ is $........$.
→If $A(l, –1, 2), B(5, 7, –6), C(3, 4, –10)$ and $D(–l, –4, –2)$ are the vertices of a quadrilateral $\text{ABCD},$ then its area is :
→The latus rectum of an ellipse is $10$ and the minor axis is equal to the distance between the foci. The equation of the ellipse is
→