The graph of f(x)=(lnx) / √(x) is shown in Fig. 3 . Find the coordinates of the maximum point.

Name: Problem 23, Chapter 4: Calculus and Its Applications
Uploaded: 2021-09-01T06:02:55-08:00
Duration: 3 min 8 s
Channel: Darshan Maheshwari
Description: The graph of f(x)=(lnx) / √(x) is shown in Fig. 3 . Find the coordinates of the maximum point.

The graph of f(x)=(lnx) / √(x) is shown in Fig. 3 . Find the...

Key Concepts

Domain Considerations

Domain considerations are crucial when dealing with functions, especially those involving logarithms or radicals. Ensuring that the function is evaluated only where it is defined prevents invalid computations and guides the search for extrema within the appropriate interval.

Optimization

Optimization involves finding the maximum or minimum values of a function. In calculus, this process typically involves computing the derivative, determining the critical points by setting the derivative equal to zero, and then using tests such as the first derivative test or the second derivative test to classify these points as maxima or minima.

Derivative

The derivative of a function represents its instantaneous rate of change with respect to its independent variable. It is a fundamental tool in calculus that allows one to analyze how a function behaves, highlighting regions where it increases, decreases, or remains constant.

Differentiation Rules

Differentiation rules, such as the product rule, chain rule, and quotient rule, are systematic methods for finding the derivative of complex functions. These rules allow decomposing complex expressions into simpler parts whose derivatives can be computed and then combined to obtain the overall derivative.

Critical Points

Critical points are values in the domain of a function where its derivative is zero or undefined. These points are important because they are potential candidates for local minima, local maxima, or points of inflection, and identifying them is a key step in the process of optimization.

Transcript

00:01 Question we are already given the graph of l and x over root x over here and we got to find the coordinates of the maximum point which looks somewhere over here so definitely we've got to take the help of the derivative so we differentiate this and find the first derivative so we've got to use the quotient rule so we have the denominator square that is root x square and we have denominator times differentiation of numerator which is natural log of x minus numerator which is lnx and then differentiation of denominator which is root x so root x remains azealous differentiation of ln x is 1 over x minus ln x remains as a certain differentiation of root x is 1 over 2 root x and a denominator root x square is just x so root x over x is 1 over root x minus uh this this this is nothing but natural log of x over 2 root x over x over x if you want to say simplify this and the denominator x remains as it is in the numerator the common denominator is 2 root x over here we have 2 minus natural log of x so this becomes 2 minus natural log of x over 2x raised 2 3 over 2 that's how it is and this is the derivative is the first derivative of the function so in order to find the maxima we equate the first derivative to so definitely this means that a denominator in a rational expression if that is equal to zero it means the numerator is equal to zero minus natural log of x is zero which means that the natural log of x is 2 just means that x is equal to e squared in order to find whether that's a maximum or minima we've got to use the first derivative test so this is e square and for any value of x less than e square this is going to be this expression rather is going to be a positive because 2 minus e squared at e squared that's 0 so for any value which is less than e squared that's going to be less than 2 so that's going to be positive and this is going to be negative so since the transition is from positive to negative so we have a local maxima at that particular point so the value of x we already got and the corresponding value of fx is going to be f x is equal to 1 minus in fact that's lnx so we root x so that's going to be ln in fact there's the value of f e square so that's ln e square over root of e square so l n e square is nothing but two times l and e because as for the properties of logs is going to drop down and root of e square is nothing but e and l ne is nothing but one so that's two or e so the coordinates of the maximum point is going to be so the final answer is value of x is e square and the corresponding value of f x is 2 over e, which is the final answer...

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