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) = x³ - 13x² - 9x

Accepted Answer

Welcome back, everyone. Locate the critical points of F of X equals 2X cubed minus 24 X2 + 72 X and use the second derivative test to identify whether these points are local, maxima, or minimum. So first of all, let's differentiate the function we want to identify F of X, which means that we're differentiating to X cubed minus 24 x 2. Plus 72 X. Here we have to apply the power rule which says that the derivative of the first term is 3 multiplied by 2, so that's 6 X squared, right? We're decreasing the exponent by 1 unit followed by negative 48 X plus 72. Now we want to identify the critical points, so we are either setting the derivative equal to 0 or checking where it does not exist. This is a polynomial, so we just want to check where this polynomial is equal to 0. What we can do is simply factor out the 6. Which gives us 6 x squared minus 8 x + 12 equals 0, and now we can factor out further, which gives us 6 multiplied by x minus 2 multiplied by x minus 6 equals 0. And as we can see now that we have a completely factored form. We have two possible solutions. X belongs to a set of 2 numbers because we have two factors involving X, so either it is 2 or 6. Right? Because for these values, the product is going to be equal to 0. And now that we have our critical points, we're going to use the second derivative test to see if they correspond to minimum or maximum values. So let's identify the second derivative. We now want to differentiate 6X2 minus 48 X plus 72. According to the power rule, we get 12 bags. -48 + 0, right? Because we have a constant. And now, what we want to do is evaluate the second derivative at each vertical point. There's no necessity to evaluate exactly. We just want to identify the sign of the second derivative. So F at 0.2 is going to be equal to 12 multiplied by 2 minus 48. So that's 24 minus 48, which is negative, right? Or basically negative 24 if we want to evaluate it. We can also notice that the negative term has a higher magnitude than the positive term. And if the second derivative is negative, what does that mean? Well, the function is concave down. So we are expecting to have a local maximum. Now F double at the second critical point, which is 6, it's going to be equal to 12 multiplied by 6 minus 48, so that's 72 minus 48, which is positive. If it is positive, this means that the function is concave up, and if the function is concave up, then this is a point of a local minimum. Well done. So now that we understand that our function has a minimum at 6 and a maximum at 2, we can say that. The minimum value of the function f min. Is obtained at 0.6 and that minimum is equal to that we want to state the value right and to get that value we need to take the original function and evaluate it at 6. So we take F of 6, which is 2 multiplied by 6 cubed minus 24 multiplied by 6 squad plus 72 multiplied by 6. And this gives us Sierra. So we are going to say the minimum value of the function is obtained at 0.6 and that minimum is equal to 0, while the maximum value of max is obtained at 0.2 X equals 2, and it is equal to. We want to evaluate F at.2 which gives us 2 multiplied by 2 cubed minus 24 multiplied by 2 squad, plus 72 multiplied by 2. And this gives us 64. So we're going to say equals 64. And that's it. We have our answers. We have 2 critical points. Their X coordinates are 2 and 6. The function obtains its local minimum at X equals 6, and that minimum is 0. The maximality of the function is 64, and it is obtained at X equals 2. 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) = x³ - 13x² - 9x

Key Concepts

Critical Points