Find the absolute maximum and minimum values of the...

Find the absolute maximum and minimum values of the following function on the given interval. If there are multiple points in a single category list the points in increasing order in x value and enter N in any blank that you don't need to use. f(x) = 6(x^2 - 1)^3, [-1, 2] Absolute maxima x = y = x = y = x = y = Absolute minima x = y = x = y = x = y =

Transcript

00:01 In this question, we are asked to find the absolute maxima and the absolute minima of the given function over the given interval.

00:08 And the only points where the absolute maxima and the absolute minima can occur are either the endpoints of the interval, in our case negative 1 and 2, or the critical points.

00:27 The critical points are the points where either f prime of x equals to 0 or f prime of x is undefined.

00:39 So we need to find f prime of x to find the critical points.

00:45 By the chain rule, f prime of x equals to 6, multiplied by the derivative of x squared minus 1, and multiplied by the derivative of the third power.

01:02 And this simplifies to 12 times 3, 36 x times x squared minus 1 squared.

01:22 Now note that this function is defined for all.

01:26 X whatever x you plug in you're going to get a nice answer so there are no critical points from f prime of x being undefined so we just need to solve the equation f prime of x equals to zero so we want to solve the equation 36x times x squared minus 1 squared equal 0 one solution is x equals 0 the other solution is x squared minus 1 squared equal 0 or simply x squared minus 1 squared 0 or x squared equals to 1 meaning that x equals to plus minus 1 so the critical the possible critical points are x equals 0 x equals negative 1 and x equals 1 now we have to be careful because we want the critical points to be inside this interval from negative 1 to 2 and all of them are inside the interval which so we're going to work with all of them now we need to calculate the values of our function, the value of our function, at the critical points and at the end points.

02:55 So we need to calculate there are three critical points, 0, negative 1 and 1, and there are 2 end points, negative 1 and 2.

03:03 And it happens that one of the critical points coincides with one of the end points.

03:10 So we need to calculate the value of our function at 4 points.

03:15 F of 0, f of negative 1, f of 1, and the end points are negative 1 and 2, but f of negative 1 is already there, so we need to calculate f of 2...

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