Write the point where f(x)=x _ e x attains minimum value. - Vidyadip

Question

Write the point where $\text{f}(\text{x})=\text{x}\log_{\text{e}}\text{x}$ attains minimum value.

✓

Answer

Given, $\text{f}(\text{x})=\text{x}\log_{\text{e}}\text{x}$
$\Rightarrow\text{f}'(\text{x})=\log_{\text{e}}\text{x}+1$
For a local maxima or a minima, we must have $\text{f}(\text{x})=0$
$\Rightarrow\log_{\text{e}}\text{x}+1=0$
$\Rightarrow\log_{\text{e}}\text{x}=-1$
$\Rightarrow\text{x}=\frac{1}{\text{e}}$
$\Rightarrow\text{f}\Big(\frac{1}{\text{e}}\Big)=\frac{1}{\text{e}}\ \log_{\text{e}}\Big(\frac{1}{\text{e}}\Big)=-\frac{1}{\text{e}}$
Now, $\text{f}''(\text{x})=\frac{1}{\text{x}}$
At, $\text{x}=\frac{1}{\text{e}}$
$\text{f}''\Big(\frac{1}{\text{e}}\Big)=\frac{1}{\frac{1}{\text{e}}}=\text{e}>0$
So, $\Big(\frac{1}{\text{e}},-\frac{1}{\text{e}}\Big)$ is a point of local minimum.

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 "2 Marks Questions" from APPLICATION OF DERIVATIVES→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 $\vec{\text{b}}$ is a unit vector such that $\big(\vec{\text{a}}+\vec{\text{b}}\big).\big(\vec{\text{a}}-\vec{\text{b}}\big)=8,$ find $|\vec{\text{a}}|.$

For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

$\text{y}=\text{a e}^{\text{x}}+\text{b e}^{-\text{x}}+\text{x}^2$

$\text{x}\frac{\text{d}^2\text{y}}{\text{dx}^2}+2\frac{\text{dy}}{\text{dx}}-\text{xy}+\text{x}^2-2=0$

Construct a $2 \times 2$ matrix $A = [a_{ij}]$ whose elements $a_{ij}$ are given by:
$\text{a}_\text{ij}=\frac{|2\text{i}-3\text{j}|}{2}$

A company produces two types of goods A and B, that require gold and silver. Each unit of type A requires 3 g of silver and 1 g of gold while that of type B requires 1 g of silver and 2 g of gold. The company can procure a maximum of 9 g of silver and 8 g of gold. If each unit of type A brings a profit of ₹ 40 and that of type B ₹ 50, formulate LPP to maximize profit.

Evaluate $\left|\begin{array}{cc} {x} & {x+1} \\ {x-1} & {x} \end{array}\right|$

If $\vec{\text{a}}.\vec{\text{a}}=0$ and $\vec{\text{a}}.\vec{\text{b}}=0,$ what can you conclude about the vector $\vec{\text{b}}?$

If A and B are independent events, where $P ( A )=\frac{1}{2}$, $P ( A \cup B )=\frac{3}{5}$ and $P ( B )=x$, then find the value of $x$.

Show that the function $f(x) = x^3 – 3x^2 + 6x – 100$ is increasing on R.

Show that the Signum Function f : R $\rightarrow$ R, given by $f(x) = \left\{ {\begin{array}{*{20}{c}} {1,\;if\;x > 0} \\ {0,\;if\;x = 0} \\ { 1,\;if\;x < 0} \end{array}} \right.$ is neither one-one nor onto.

Integrate the function: $\frac{1}{{x - \sqrt x }}$