9-18 (a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum a

9-18 (a) Find the intervals on which f is increasing or...

Key Concepts

First Derivative Analysis

This concept involves computing the first derivative of a function to determine its rate of change. By analyzing where this derivative is positive or negative, we identify the intervals over which the function is increasing or decreasing. This method is crucial for understanding the behavior of a function's graph and locating local extremes.

Critical Points and Local Extrema

Critical points are the values of the domain where the first derivative is zero or undefined. These points are potential candidates for local maximum or minimum values. By examining each critical point with tests such as the first or second derivative test, one can determine the presence and nature of local extrema.

Second Derivative and Concavity

The second derivative of a function provides information about the concavity of the function’s graph. If the second derivative is positive over an interval, the graph is concave upward; if negative, it is concave downward. This analysis is essential for understanding the curvature of the function.

Inflection Points

Inflection points occur where the function changes its concavity. They are identified by points where the second derivative changes sign. Recognizing inflection points helps in accurately sketching the graph of the function and understanding its curvature transitions.

Transcript

00:01 We're given a function f with formula f of x equals x to the fourth times e to the negative x.

00:17 In part a, we're asked to find the intervals on which f is increasing or increasing.

00:23 F prime of x is equal to using the product rule, x to the fourth times negative e to the negative x plus 4x cubed, e to the negative x.

00:40 And after factoring, we get x cubed e to the negative x times 4 minus x.

01:01 So it follows that f prime of x, the derivative, is greater than zero.

01:09 If x has to be less than negative 4, you should just write this differently, the same as negative x cubed, e to the negative x times x minus 4.

02:00 So if x is less than negative 4, this is going to be negative.

02:07 This will be negative.

02:09 It needs to be negative, so be negative.

02:11 Ok, if x lies between negative 4 and 0, then x cubed will be negative, and x minus 4 will be positive.

02:26 So this will be a negative times a negative, which is positive.

02:32 And likewise, we have the f double prime of x is less than 0 if x is less than negative 4, or if x is greater than 0.

02:55 Sorry, this is wrong.

02:57 This should be if x is between 0 and 4.

03:05 And then f double prime of x should be just f prime of x is less than 0, if x is less than 0, if x is less than 0, or if x is greater than 4.

03:23 So we have that f is increasing on the open interval 0 to 4, and that f is decreasing on the open intervals negative infinity 0 and 4 infinity.

03:44 In part b, we're asked to find the local max and minimum values of f.

03:54 We see the f changes from decreasing to increasing at x equals 0.

04:11 And f changes from increasing to decreasing at x equals 4.

04:21 So f of 0, which if we plug this into our formula for f is zero, this is a local minimum value of f.

04:37 And f of 4, we plug this into our formula for f, is 256 over e of the 4.

04:56 A local maximum value of f.

05:06 Finally, in part c, we're asked to find the intervals of concavity and the points of inflection of f...

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