Download App

6. Application of derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths6. Application of derivatives1 Mark
MCQ
The function ${x^x}$ is increasing, when
  • ✓
    $x > {1 \over e}$
  • B
    $x < {1 \over e}$
  • C
    $x < 0$
  • D
    For all real $ x$
✓

Answer

Correct option: A.
$x > {1 \over e}$
a
(a) Let $y = {x^x}$ ==> $\frac{{dy}}{{dx}} = {x^x}(1 + \log x)$

For $\frac{{dy}}{{dx}} > 0$;

${x^x}(1 + \log x) > 0$ ==> $1 + \log x > 0 \Rightarrow {\log _e}x > {\log _e}\frac{1}{e}$

For this to be positive, $x$ should be greater than $\frac{1}{e}$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from 6. Application of derivatives→6. Application of derivatives — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $\int\limits_0^2 {} \,375 \,x^5 (1 + x^2)^{-4}\, dx = 2^n$ then the value of $n$ is :
→
The number of bijective functions $f :\{1,3,5, 7, \ldots \ldots . .99\} \rightarrow\{2,4,6,8, \ldots \ldots, 100\}$, such that $f(3) \geq f(9) \geq f(15) \geq f(21) \geq \ldots \ldots f(99), \quad$ is
→
If the area bounded by $y = a{x^2}$ and $x = a{y^2}$, $a > 0$, is $1$, then $a = $
→
Let $M$ and $N$ be two $3 \times 3$ matrices such that $M N=N M$. Further, if $M \neq N^2$ and $M^2=N^4$, then

$(A)$ determinant of $\left( M ^2+ MN ^2\right)$ is $0$

$(B)$ there is a $3 \times 3$ non-zero matrix $U$ such that $\left( M ^2+ MN ^2\right) U$ is the zero matrix

$(C)$ determinant of $\left( M ^2+ MN ^2\right) \geq 1$

$(D)$ for a $3 \times 3$ matrix $U$, if $\left( M ^2+ MN ^2\right) U$ equals the zero matrix then $U$ is the zero matrix

→
Two matrices of same order are said to be equal if the ______ of the two matrices are equal.
→
If $y(\theta)=\frac{2 \cos \theta+\cos 2 \theta}{\cos 3 \theta+4 \cos 2 \theta+5 \cos \theta+2}$ then at $\theta=\frac{\pi}{2}, y^{\prime \prime}+y^{\prime}+y$ is equal to:
→
On the interval $\left( {0,{\pi \over 2}} \right)$, the function $ log \,sin \,x $ is
→
Choose the correct answer from the given four options.
The matrix $\begin{bmatrix}0&-5&8\\5&0&12\\-8&-12&0\end{bmatrix}$ is a:
  1. Diagonal matrix.
  2. Symmetric matrix.
  3. Skew-symmetric matrix.
  4. Scalar matrix.
→
$\text{The value of}\ \hat{\text{i}}\cdot(\hat{\text{j}}\times\hat{\text{k}})+\hat{\text{j}}\cdot(\hat{\text{i}}\times\hat{\text{k}})+\hat{\text{k}}\cdot(\hat{\text{i}}\times\hat{\text{j}})\ \text{is}$
  1. 0
  2. -1
  3. 1
  4. 3
→
The Convex Polygon Theorem states that the optimum (maximum or minimum) solution of a LPP is attained at atleastone of the ______ of the convex set over which the solution is feasible.
→