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) = 2x⁻³ - x⁻²

Accepted Answer

Welcome back, everyone. Locate the critical points of F of X equals 3x the power of -4 minus X to the power of -3, and use the second derivative test to identify whether these points are local, maxima, or minimum. So basically what we want to do is to differentiate the function to identify its critical points. We want to identify F of X. And this means that we are differentiating. We ask the power of -4. Minus X the power of -3. So now we want to apply the power rule. -4 multiplied by 3 gives us -12, followed by X, the power of -4 minus 1, which is -5 minus that we have to bring down -3, so negative of negative gives us a positive sign. 3 X to the power of -3 minus 1, which is -4. We have our first derivative, we can actually rewrite it as -12 divided by X, the power of fifth, using the reciprocal rule of fractions. Plus we divided by X the power of Y. What we want to do is simply set it to 0 or check where it does not exist. Well, it does not exist at X equals 0 because division by 0 is undefined, but this point is not in the domain of the function because it's also undefined at 0.0. So we basically want to set it to 0, and this will be the only condition that we want to check. What we want to do is simplifying the common denominator, so we're going to have -12. Plus 3 X because we need to multiply and divide the second term by X to get the common denominator of X to the power of 5. And now we're setting it to 0. If our rational function is equal to 0, we want to make sure that our numerator is equal to 0, so we're saying -12 + 3 X is equal to 0, and our denominator is not equal to 0, right? So now solving the first equation, we can rearrange it and we can say that 3 X is equal to 12, which means that X is equal to 4. And this is the critical point that we have, right? We want to classify that critical point using the 2nd derivative thus, so let's identify the second derivative. And this means that we are performing differentiation on -12 X the power of -5, plus 3x the power of -4. Applying the power rule, we're going to get 60 X to the power of -6, minus 12 X to the power of -5. We can rewrite it as 60 divided by X. To the 6 minus 12 divided by X the power of 5. Once again, we can find the common denominator. So we're going to get 60 minus 12 x divided by X to the power of 6. Right? Because once gained for the 2nd term, we are dividing and multiplying by X to get the common denominator of X to the power of 6. And now what we want to do is simply evaluate the second derivative at the critical point and see if we get a positive or a negative sign, or maybe it's 0, right? So F add 4 is going to be equal to 60 minus 12 multiplied by 4 divided by. Or to the power of sex. Evaluating the numerator, we get 60 minus 48. So that's 12, and we're dividing it by a positive number, which is a positive ratio. So there's no necessity to identify the actual value if we can understand that this is a positive number. We will be able to also understand that our function is concave up at the critical point, and if it is concave up, this means that we have a local minimum. Now, we understand from the second derivative test that X equals 4 is a local minimum point, and we simply want to identify the value of the function at that point. So we are evaluating F at 0.4. Which gives us 3 x to the power of -4 or basically 3 divided by 4 to the power of 4 minus x to the power of -3 or basically 1 divided by Xq, right? So our axis 4, we are taking 1 divided by 4 to the power of 3. Let's evaluate the result. Once again, we can identify the common denominator. So that would be 4 to the power of 4. And in our numerator we get 3 minus 4. This gives us -1 divided by. Or to the power of 4, so 16 squared, right, which is 256. And this means that we can report our answer which is The value of the function where it has a minimum. Is going to be equal to -1. Divided by 256, and the X coordinate for the minimum is 4. So F min at 0.4 equals -100, 1 divided by 256. Basically this means that. The minimum value of the function F is equal to -1 divided by 256, and that minimum value is obtained when x equals 4. This is what the notation represents. Let's label it as our final answer, and thank you for watching.

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) = 2x⁻³ - x⁻²

Key Concepts

Critical Points