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 the function $f(x) = \frac{x}{2} + \frac{2}{x}$.To find the local extrema,we first find the derivative $f'(x)$:$f'(x) = \frac{1}{2} - \frac{

Vedclass · Accepted Answer

$ 2 $

Vedclass · Answer

(D) Given the function $f(x) = \frac{x}{2} + \frac{2}{x}$.To find the local extrema,we first find the derivative $f'(x)$:$f'(x) = \frac{1}{2} - \frac{2}{x^2}$.Setting $f'(x) = 0$ to find critical points:$\frac{1}{2} - \frac{2}{x^2} = 0 \Rightarrow \frac{2}{x^2} = \frac{1}{2} \Rightarrow x^2 = 4 \Rightarrow x = 2, -2$.Now,we find the second derivative $f''(x)$ to test for local minima:$f''(x) = \frac{d}{dx}(\frac{1}{2} - 2x^{-2}) = 0 - 2(-2)x^{-3} = \frac{4}{x^3}$.Evaluating 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 $x = 2$.

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

Similar Questions

If $f(x) = \sin x - x \cos x$ has a maximum at $x = n\pi$,then which of the following is true?

If the absolute maximum value of the function $f(x) = (x^2 - 2x + 7) e^{(4x^3 - 12x^2 - 180x + 31)}$ in the interval $[-3, 0]$ is $f(\alpha)$,then:

The sixth term of an $A.P.$ is equal to $2$. The value of the common difference $x$ of the $A.P.$ that makes the product $a_1 a_4 a_5$ least is given by:

The maximum value of $x e^{-x}$ is

The function $f(x) = x^3 - 4x^2 + 4x + 3$ defined on $[-1, 3]$ has

Vedclass Test Series

Exam Paper Generator

Online Exam Module