80-81 Draw a graph of f that shows all the important aspects of the curve. Estimate the local maximum and minimum values and then use calculus to find these values exactly. Use a graph of f to estimate the inflection points. f(x)=ex3x
To find: The local maximum and local minimum by drawing a graph of the given function and then estimate the inflection points by using the graph of .
The local minimum and local maximum values of the given function are and 1.5 respectively.
The inflection points of the function are
You may know:
Local Maxima and Local Minima:
If is any real valued function and c be an interior point in the domain of . Then,
(i) The point c is called a point of local maxima if there is an such that,
(ii) The point c is called a point of local minima if there is an such that,
The inflection points are those point at which the second derivative of the function is zero.
The exponential function,
The given function is .
Differentiate the given function with respect to x.
Find the critical point of the function by equating the above derivative to zero.
Find the intercepts of the graph of the function.
Substitute in to find the y-intercept.
So, the y-intercept is
Also, there is no x-intercept.
It can be noted that when x tends to negative infinity, the y-value tends to 0.
So, the x-axis, that is, is the asymptote to the graph of the given function.
The graph of the given function is shown below:
From the above graph, it can be noted that the local maximum occurs at the point and local minimum occurs at the point .
The local minimum and local maximum values can be found as follows:
Calculation of Local Maximum and Local Minimum using Calculus:
Find the second derivative of the given function.
It can be observed that,
This shows that the local maximum occurs at the point and local minimum occurs at the point .
To find the inflection point, draw the graph of .
Final statement: the inflection points of the function are .