Second Derivative Test Locate the critical points of the follo...

Question

Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.

f(x) = x²e⁻ˣ

Accepted Answer

Below there, today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Locate the critical points of F of X is equal to X the power of 3, multiplied by E power of -2 X. And use the second derivative test to identify whether these points are local maxima, minima, or neither. Awesome. So it appears for this particular problem we're asked to solve for three separate answers. Essentially, we're trying to locate the critical points of this function F of X, and we're asked to use the second derivative test to help us identify whether these points are local maxima, minima, or neither values. So we need to locate the critical points and once we've located the critical points, we can figure out. If these points are local maximum, minima, or neither, and that will be our final answer, or rather answers that we're trying to solve for. So with that said, First off, we need to consider the function F of X is equal to x 3 multiplied by E of -2 X, and we need to recall a note that its critical points are going to be located at. And let's make a note of this, they're gonna be located at 0.0. And 3 divided by 2. 27 divided by 8 multiplied by E to the power of 3, and we will need to use the second derivative test in order to identify whether these points are local maxima, minima, or neither. Awesome. So with that said, we must first, our first step is we need to determine the derivative of F of X. So the first derivative of F of X. So we need to say F of X is going to be equal to drumroll D by DX of X 3 multiplied by E the power of negative to X. Fantastic. So when we perform the derivative, we will find that F of X is going to be equal to E power of -2X multiplied by 3 x 3 minus 1 plus X 3 multiplied by E of -2 X multiplied by -2. And when we simplify this expression down to its most simple form, we will find that our derivative for F of X, so F of X, is going to be ultimately equal to E to the power of -2X multiplied by 3 x 2 minus 2 x 3. And to note, make a quick note that I skipped a few steps, but ultimately you're just using algebra to simplify your expression for F of X down to its most simple form, or you can't simplify anymore. Awesome. So now we need to determine the second derivative of F of X. So we need to find F of X and F of X is going to be equal to D by DX of E to the power of -2X multiplied by 3 x 2 minus 2x the power of 3. Fantastic. And in order to do this, we must recall and use the product rule for differentiation, which states that D by DX of U multiplied by V is equal to V multiplied by U plus U multiplied by V. So with that in mind, we could write that F of X is equal to 3 x 2 minus 2 x 3, multiplied by E-2X multiplied by -2 plus E of -2X multiplied by 6 X minus 6 X to the power of 2. Fantastic. And once again, I'm gonna skip a few steps because you're trying to simplify this expression down to its most simple form. And when you do that, you will find that F of X is going to be equal to 2 X. Multiplied by E, so it's gonna be 2 X multiplied by E to the power of negative 2 X multiplied by 3 minus 6 X. So 3 minus 6 X plus 2 X to the power of 2. And once again, that is our most simplified form of F of X. Fantastic. So now, Our next step, and let's say this as step 3, is we now need to evaluate F of X at these specific critical points. So we need to evaluate it at F of X is greater than 0. So when F of X is greater than 0, then F of X will have a local minimum. So I'll call it Emin, and Emin. I'll do L hyphenin. So, a local minimum at X equals C, and that will mean that the function will be concave up, and if F of X is less than 0, then that will mean that we will have a local maximum. At So I have a local maximum at X equals C, and the function will be concave down. And if F of X is equal to 0, that will mean that the test will be inconclusive. And if it's inconclusive, that will mean that further analysis, such as the first derivative test will be needed. So with that said, we need to note that when X is equal to 0, that will mean that F of 0, because you're plugging in a value of 0 in for X. That will mean that it'll be equal to 2, multiplied by 0, multiplied by E to the power of -2 multiplied by 0, multiplied by 3 minus 6 multiplied by 0. Plus 2 multiplied by 0 to the power of 2, which will be all equal to 0. And because F of 0 is equal to 0, that means that the second derivative test is inconclusive. So this will mean that it's inconclusive. So that means that we need to use the first derivative test at X is equal to 0, so we need to check the sign of F of X is equal to x2 multiplied by the power of -2 X multiplied by 3 minus 2 X at around X equals 0. So for when X is less than 0, let us say that X is equal to -1. So F of -1 is going to be equal to -1 to 2, multiplied by the power of -2 multiplied by -1, multiplied by 3 minus 2 multiplied by -1. And when we evaluate this, it will be equal to 5 multiplied by E to the power of 2. Fantastic. And for when F of X is greater than 0. So that will mean that when F of X is greater than 0. So for when X is greater than 0, so now we're considering X is greater than 0, let us say that X is equal to 1, so we can go ahead and write that F of 1 is equal to 1. Squared multiplied by e to the power of -2 multiplied by 1, multiplied by 3 minus 2 multiplied by 1, which will be equal to 1 divided by E to the power of 2, and this will be for the condition where F of X is greater than 0. And since F of X does not change it like change signs at X equals 0, that will mean that there will be no local maximum or minimum at 0.0. So, let's make a note of that. So no Max or Min. At 0.0. Fantastic. So, once again, to reiterate, and let's put a box around that since that's one of the answers that we're trying to solve for. So since, once again, since F of X does not change change sign at X equals 0, there will be no local maximum or minimum at 0.0 in parentheses. So for when X is equal to 3 halves or 3 divided by 2, We will determine that F of, and note that we need to note that this is not just F of anything, it's gonna be F of 3 divided by 2 Bs we're plugging in 3 divided by 2 in for X. So that will mean it'll be equal to 2 multiplied by 3 divided by 2, and 2 multiplied by 3 divided by 2 multiplied by E to the power of -2 multiplied by 3 divided by 2. Multiplied by 3, minus 6 multiplied by 3 divided by 2. Plus, 2 multiplied by 3 divided by 2. And don't forget that that value was squared. Fantastic. So, Essentially, what we're going to be doing is we're going to be evaluating this expression, and we're gonna be trying to simplify it down to its most simple form, and when we do, we will find that F of 3 halves or 3 divided by 2 is going to be equal to -9 divided by 2 multiplied by E to the power of 3. And this will be holding true for the condition where F of 3 divided by 2 is less than 0. So, therefore, there will be a local maximum, so local maximum at 3 divided by 23 halves, 27 divided by 8 multiplied by E to the power of 3. Fantastic, we did it. We've solved for this problem. Hooray. So going back to look at our multiple choice answers to see which answer corresponds with what we determined together. And ultimately, without a doubt, our final answer, it's important to note that it's gonna be neither. Maximum Nor minimum, so it's gonna be neither maximum nor minimum. At 0.0 in parentheses. So that is one of our final answers, going quickly recapping here together what we determined together, so this is a quick recap. So that's one of our answers. And secondly, we need to know that there's going to be a local maximum at Parentheses 3 divided by 2, 27 divided by or it's gonna be 27 divided by. 8 E to the power of 3, and that's it. Those are our last two answers. Hooray, we did it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.

f(x) = x²e⁻ˣ

Key Concepts

Critical Points