The function f(x)=(x)/(2)+(2)/(x) has a local minimum at (A) x=2 (B) x=-2 (C) x=0 (D) x=1

Name: Problem 169, Chapter 14: A Complete Resource Book in Mathematics for JEE Main
Uploaded: 2021-09-20T03:11:39-08:00
Duration: 1 min 57 s
Channel: Mahipal Kumawat
Description: The function f(x)=(x)/(2)+(2)/(x) has a local minimum at (A) x=2 (B) x=-2 (C) x=0 (D) x=1

step 1 The question given function fx is equal to x by 2 plus 2 by x now take the differentiation of th

The function f(x)=(x)/(2)+(2)/(x) has a local minimum at (A)...

Key Concepts

Function Domain Analysis

Analyzing the domain of a function is essential before applying optimization techniques. It ensures that the points being considered, especially those derived from setting the derivative to zero, are valid within the context of the function's definition, avoiding points where the function might be undefined.

Derivative

The derivative of a function measures its rate of change and is a fundamental tool in calculus for determining where a function is increasing or decreasing. By computing the derivative, one can find points in the domain where the slope is zero or undefined, which are often potential candidates for local maxima or minima.

Critical Points

Critical points are values in the domain of a function where its derivative is either zero or undefined. These points are essential in the study of calculus as they are potential locations for local extreme values (maxima or minima), warranting further investigation using derivative tests.

First Derivative Test

The first derivative test is used to determine the local extreme nature of a function at its critical points. By examining the sign changes of the derivative before and after a critical point, one can conclude whether the function attains a local minimum (derivative changes from negative to positive), a local maximum (positive to negative), or neither.

Second Derivative Test

The second derivative test provides an alternative method for classifying critical points. By evaluating the second derivative at a critical point, if the value is positive, the function is concave up and has a local minimum at that point; if negative, the function is concave down and has a local maximum. This test also gives information about the curvature of the function near the point.

Transcript

Please give Ace some feedback