The function f(x) = frac{x}{2} + frac{2}{x} h | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given function is $f(x) = \frac{x}{2} + \frac{2}{x}$.First,find the derivative $f'(x)$:$f'(x) = \frac{1}{2} - \frac{2}{x^2}$.For local maxima or m

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(D) Given function is $f(x) = \frac{x}{2} + \frac{2}{x}$.First,find the derivative $f'(x)$:$f'(x) = \frac{1}{2} - \frac{2}{x^2}$.For local maxima or minima,set $f'(x) = 0$:$\frac{1}{2} - \frac{2}{x^2} = 0 \implies \frac{1}{2} = \frac{2}{x^2} \implies x^2 = 4 \implies x = \pm 2$.Now,find the second derivative $f''(x)$:$f''(x) = \frac{d}{dx}(\frac{1}{2} - 2x^{-2}) = 0 - 2(-2)x^{-3} = \frac{4}{x^3}$.Check the sign of $f''(x)$ at critical points:At $x = 2$,$f''(2) = \frac{4}{2^3} = \frac{4}{8} = \frac{1}{2} > 0$.Since $f''(2) > 0$,the function has a local minimum at $x = 2$.At $x = -2$,$f''(-2) = \frac{4}{(-2)^3} = \frac{4}{-8} = -\frac{1}{2} Since $f''(-2) < 0$,the function has a local maximum at $x = -2$.

The function $f(x) = \frac{x}{2} + \frac{2}{x}$ has a local minimum at $x = $ ..........

Similar Questions

Which one of the following is not a functional unit of an ecosystem?

$A$ wire in the form of a square of side $a$ carries a current $i$. Then,the magnetic induction at the centre of the square is (Magnetic permeability of free space $= \mu_0$)

$A$ frictionless track $ABCDE$ ends in a circular loop of radius $R$. $A$ body slides down the track from point $A$ which is at a height $h = 5 \, cm$. The maximum value of $R$ for the body to successfully complete the loop is .................. $cm$.

In Gattermann reaction,a diazonium group is replaced by $\underline{X}$ using $\underline{Y}$. $\underline{X}$ and $\underline{Y}$ are:
$\underline{X} \quad \underline{Y}$

$A$ Young's double slit experiment uses a monochromatic source. The shape of the interference fringes formed on a screen is

Vedclass Test Series

Exam Paper Generator

Online Exam Module