Write the minimum value of f(x) = x^x . - Vidyadip

Question

Write the minimum value of $f(x) = x^x .$

✓

Answer

We have $\text{f}(\text{x})=\text{x}^{\text{x}}$
$\therefore\ \text{f}'(\text{x})=\text{x}^{\text{x}}(\log\text{x}+1)$
For maxima and minima, $f'(x) = 0$
$\Rightarrow \text{x}^{\text{x}}(\log\text{x}+1)=0$
$\Rightarrow \text{x}=\text{e}^{-1}$
Now,
$\therefore\ \text{f}''(\text{x})=\text{x}^{\text{x}}(\log\text{x}+1)^{2}+\frac{\text{x}^{\text{x}}}{\text{x}}$
At $\text{x}=\frac{1}{\text{e}}$
$\therefore\ \text{f}''(\text{x})>0\ \text{as}\ \text{x}^{\text{x}}(\log\text{x}+1)^{2}+\frac{\text{x}^{\text{3}}}{\text{x}}>0$
$\therefore \text{x}=\frac{1}{\text{e}}$ is the point of local minima.
Hence, minimum value $=\text{f}\Big(\frac{1}{\text{e}}\Big)=\Big(\frac{1}{\text{e}}\Big)^\frac{1}{\text{e}}=\text{e}^\frac{-1}{\text{e}}$ .

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 "5 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

Evaluate:
$\DeclareMathOperator*{\median}{\text{lim}} \median_{\text{x}\rightarrow0}\frac{\text{tan x - sin x}}{\sin^{3}\text{x}}$.

Differentiate the following functions with respect to x:
$\sin^{-1}\Big\{\sqrt{\frac{1-\text{x}}{2}}\Big\},0<\text{x}<1$

Prove that $x^2 - y^2 = c(x^2 + y^2)^2$ is the general solution of differential equation $(x^3-3xy^2)dx=(y^3-3x^2y)dy$, where c is a parameter.

Evaluate the following integrals:
$\int\frac{1}{(\text{x}^2-1)\sqrt{\text{x}^2+1}}\text{ dx}$

If $y = 2 \cos (\log x) + 3 \sin (\log x),$ Prove that $x^{2} \frac{d^{2}y}{dx}^{2} + x\frac{dy}{dx} + y = 0.$

Show that the differential equation $2ye^{x/y} dx + (y - 2xe^{x/y} ) dy = 0$ is homogeneous. Find the particular solution of this differential equation, given that $x = 0$ when $y = 1.$

Find A and B so that $\text{y}=\text{A}\sin3\text{x}+\text{B}\cos3\text{x}$ satisfy the equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}+4\frac{\text{dy}}{\text{dx}}+3\text{y}=10\cos3\text{x}.$

Two numbers are selected at random (without replacement) from positive integers 2, 3, 4, 5, 6 and 7. Let X denote the larger of the two numbers obtained. Find the mean and variance of the probability distribution of X.

$\text{Let}\ \vec{\text{a}}=4\hat{\text{i}}+5\hat{\text{j}}-\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}}$ and $\vec{\text{c}}=3\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}$ Find a vector $\vec{\text{d}}$ which is perpendicular to both $\vec{\text{c}}\ \text{and }\vec{\text{b}}\ \text{and}\ \vec{\text{d}}\cdot\vec{\text{a}}=21.$

Differentiate the functions given in Exercise:
$\text{x}^{\sin\text{x}}+(\sin\text{x})^{\cos\text{x}}$