Find the exact global maximum and minimum values of the...

Key Concepts

Exponential Functions

Exponential functions, characterized by a constant base raised to a variable exponent, have distinct growth or decay behaviors. Understanding their properties is crucial when dealing with functions that combine terms like e^x and e^(-x), as these behaviors influence the overall shape and extremal values of the function.

Second Derivative Test

The second derivative test is used to determine the concavity of the function at a critical point. By evaluating the second derivative at a critical point, one can classify the point as a local minimum if the second derivative is positive, or a local maximum if it is negative, assuming the second derivative exists.

Derivative and the First Derivative Test

The first derivative of a function provides insight into its increasing or decreasing behavior. Setting the derivative equal to zero helps identify critical points, and analyzing the sign changes of the derivative around these points can indicate whether they correspond to local minima or maxima.

Critical Point Analysis

Critical points are values in the domain of a function at which its derivative is zero or undefined. These points are important candidates for local extrema because the function’s rate of change is momentarily stationary. Analyzing these points is a fundamental step when searching for potential maximum or minimum values.

Global Extremum

This concept refers to the absolute highest and lowest values that a function attains over its entire domain. Finding a global maximum or minimum involves not only checking interior points where the function’s derivative is zero or undefined but also considering the function’s behavior at the boundaries of its domain, if any.

Transcript

00:01 We have our function f of x, and f of x is equal to 2 e to the x plus 3e to the negative x.

00:12 So we're told that the domain is all real numbers unless otherwise specified.

00:17 So the question is we wish to find the exact global maximum and or minimum of this function.

00:25 So to find global maximum in, we need to first find the derivative of this function.

00:34 So that would be f prime of x, and the derivative is 2 times e to the x, and then this will be minus 3 times e to the negative x.

00:47 Okay, so now since e is a continuous function everywhere, f prime of x exists everywhere.

00:57 So which means the only critical values that we will be able to find would be when we set f prime of x equal to zero.

01:04 Okay, so that would be crit points when f prime is equal to zero.

01:14 So we would say this is zero is equal to 2 e to the x minus 3e the negative x.

01:24 Okay, so now we just need to go ahead and solve this.

01:27 So there are different ways you can say, okay 2 e to the x is equal to 3 e to negative x and then we can multiply e to the x on both sides and we get 3 is equal to 2 e to 2x or you can think of as 3 over 2 is e to the 2x take the natural logarithm of both sides and we get 1 of 3 over 2 is equal to 2x which means x is equal to one half one of three over two.

02:07 Okay, so this is our critical point.

02:12 So now to find if this is a minimum or a maximum, you can either do the second derivative test or you can do your interval test.

02:20 So for this question, it might be a bit easier to actually do an interval test.

02:26 So what that means is you draw a number line and then you plot all your critical points on there.

02:35 So since there's only one, i'm just going to plot a point here and say this is x is equal to half of one, three over two.

02:43 And it would be beneficial for you to actually know roughly what that number is.

02:47 So let's go ahead and put that into our calculator.

02:50 So this is roughly 0 .0 .2.

02:58 Okay.

03:00 So what this means is now we want to do a test point.

03:03 Okay.

03:03 So we want to do a test point.

03:05 To the left of this critical point and then to the right of this critical point.

03:08 Critical point.

03:10 So the easiest test point would be, let's test zero.

03:13 So what is f prime of zero? well, f prime of zero is going to be two times b to the zero, which is just one, minus three times beat of the zero, which is also just one.

03:26 So this is equal to negative one.

03:28 So therefore, we have negative slope here.

03:33 Okay, and now we do a test point on the other side.

03:36 So let's say we pick one.

03:37 So f prime of 1, that would be 2e minus 3e to the negative 1.

03:46 Okay, so let's go ahead and find what that is, minus 3e to the negative 1.

03:55 Okay, so we get this is roughly 4 .3.

04:01 Okay, so that's a positive number, which means this has positive slope...