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

Second Derivative Analysis

The second derivative of a function provides insight into the concavity of the function—whether it curves upward or downward—and helps in identifying potential inflection points, where the concavity changes. By determining the sign of the second derivative over different intervals, one can deduce where the function is concave up or concave down.

Monotonicity Analysis

Monotonicity deals with determining the intervals where the function is strictly increasing or decreasing. By analyzing the sign of the first derivative, one can partition the domain into intervals where the derivative is positive (indicating an increasing function) and where it is negative (indicating a decreasing function). This analysis is crucial for understanding the overall behavior of the function and for identifying local extrema.

Concavity and Inflection Points

Concavity describes the curvature of the function: a function is concave up if its second derivative is positive and concave down if the second derivative is negative. Inflection points are those points where the concavity changes from up to down or vice versa—a formal point where the second derivative changes sign. This concept helps to provide a deeper understanding of the function's geometry and the behavior of its rate of change.

Derivative Analysis

This concept involves taking the derivative of a function to understand its behavior, such as rates of change and identifying critical points. In problems like this, one computes the first derivative to examine where the function is increasing or decreasing and to locate potential local extremum points. The process of differentiation highlights how the function's slope behaves over its domain, forming the foundation for further analysis.

Critical Points

Critical points are values in the domain of the function where its first derivative is zero or undefined. These points are of interest because they serve as candidates for local maximums or minimums. In the context of differentiable functions like the one given, finding the critical points allows us to determine where the function's increasing/decreasing behavior might change.

Transcript

00:01 We're given a function f, which has the formula f of x equals e to the 2x plus e to the negative x.

00:13 In part a, we're asked to find the intervals on which f is increasing or decreasing.

00:22 To do this, first let's find the derivative of f, f prime of x, which is 2e to the 2x minus e to the negative x.

00:31 We see that this is going to be greater than 0.

00:40 If and only if, we have that 2e to 2x is greater than e to the negative x.

00:49 And this is the same as if and only if, either the negative, or sorry, e to the positive 3x, is greater than 1 half, which is true if and only if, 3x is greater than the natural log of 2, the natural log of 1 half, which is true if and only if x is greater than 1 third of natural log of 1 minus the natural log of 2 using log properties, and this is equal to negative 1 3rd natural log of 2.

01:45 And if you must approximate this, this is about equal to negative .23.

01:55 Likewise, we have the f prime of x is less than zero if the opposite is true, so x is less than negative one -third natural log of 2.

02:19 And therefore, it follows that the function f is increasing on the open interval negative 1 3rd natural log of 2 to infinity, and f is decreasing on the open interval negative infinity, negative one -third natural log of 2.

02:44 In part b, we're asked to find the local maximum and minimum values of f.

02:52 To do so, consider when f changes from decreasing to increasing and vice versa, we see that f changes from decreasing to increasing at x equals negative one -third, natural.

03:13 Log of 2, and it never changes from increasing to decreasing...

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