The graph of f(x)=x /(lnx+x) is shown in Fig. 4 . Find the coordinates of the minimum point.

Name: Problem 24, Chapter 4: Calculus and Its Applications
Uploaded: 2021-09-01T06:06:56-08:00
Duration: 2 min 47 s
Channel: Darshan Maheshwari
Description: The graph of f(x)=x /(lnx+x) is shown in Fig. 4 . Find the coordinates of the minimum point.

step 1 This question, the graph of x over LNX plus X is given. So we need to find the coordinates of th

The graph of f(x)=x /(lnx+x) is shown in Fig. 4 . Find the...

Key Concepts

Optimization

Optimization is the process of finding the best solution from all feasible solutions. In calculus, it often involves determining the maximum or minimum values of a function. This is typically done by identifying points where the derivative is zero or undefined and then testing these points to see if they correspond to local or global extrema.

Critical Points

Critical points are the values in the domain of a function where the derivative is zero or does not exist. These points are important because they often signal potential locations for local maxima, minima, or inflection points, and they form the basis for further analysis in optimization problems.

First Derivative Test

The First Derivative Test is a method used to determine the nature of critical points. By analyzing the sign changes of the first derivative around a critical point, one can identify whether the point is a local maximum, local minimum, or neither. This test is a fundamental tool in identifying the behavior of functions around points of interest.

Differentiation Techniques

Differentiation techniques, such as the quotient rule, product rule, and chain rule, are essential for computing the derivatives of functions, especially those that are composed of multiple parts like rational or composite functions. Understanding these techniques is crucial for analyzing and solving optimization problems accurately.

Properties of Logarithmic Functions

Logarithmic functions have unique properties and differentiation rules that are important when they appear in more complex expressions. Knowing how to differentiate a logarithmic function, including its domain and asymptotic behavior, is key when it is part of a function that needs to be optimized or analyzed for extrema.

Transcript

00:01 This question, the graph of x over lnx plus x is given.

00:04 So we need to find the coordinates of the minimum point.

00:06 So it looks something like this over here, but definitely we've got to use derivative, the property of the derivative.

00:13 So the first derivative is going to look like we've got to use the question root.

00:18 So we have the denominator square times the denominator.

00:24 We have denominator times differentiation of numerator, minus the numerator which is x times the differentiation of the denominator which is ln x plus x so that's going to be x plus ln x and the differentiation of x is just one minus x remains as it is and this is a differentiation of natural log of x is 1 over x and differentiation of x is just 1 over x plus lnx whole square so this becomes x plus ln x minus if we open up the parentheses x times 1 over x is just 1 and x x times 1 is just x so here we have x over x plus ln x whole square so if we simplify this this x and minus x is cancer and we have lnx minus 1 over here over 1 plus over x plus lnx whole square so in order to find the critical points we equate the derivative to 0 so we have f dash x equated to 0 this means that whenever a rational expression is equal equated to zero, this means that the numitor must be zero.

01:36 So we have lnx minus one equated to zero, which means that lnx is equal to one.

01:41 Taking antelaw, both sides, we have the value of x as e.

01:45 Using the first derivative test, let's try to find whether this has a maximum or minima.

01:49 So at e, the sign of the derivative, if it is less than e, then this value is going to be less than one...

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