Find the absolute maximum and absolute minimum values f(x) =...

Transcript

00:01 This problem we're asked to find the absolute minimum and maximum values of this function f on the interval from 0 to 4.

00:06 And we're going to use the derivative of f to figure out when f is increasing and decreasing.

00:13 So first let's find the derivative of f f prime of x.

00:18 We can use the quotient rule to calculate this.

00:21 So we'll have the derivative of the top function, which is just one times the bottom one.

00:28 And then minus the top times the derivative of the bottom one.

00:35 And that's all over the bottom function squared.

00:43 And this simplifies to 1 minus x squared.

00:47 We have the same bottom function.

00:52 And then actually i can factor the numerator.

00:55 This will come in handy in a moment.

00:57 So 1 minus x and then 1 plus x again all over the same denominator.

01:08 And we want to set this equal to 0 because that will tell us where the critical values are.

01:15 Notice that if a fraction is equal to 0 then that means its numerator is equal to 0.

01:21 So this implies that 1 minus x times 1 plus x.

01:32 This is equal to 0.

01:33 So in particular x is equal to negative 1 and then 1.

01:38 However we're in this interval from 0 to 4.

01:42 So we don't care about negative 1.

01:44 And on this interval from 0 to 4 we have that one critical value.

01:54 1.

01:56 This figure is not to scale but i'm going to plot the sine of f prime of x.

02:06 All right...

Find the absolute maximum and absolute minimum values f(x) = 2x^3 - 3x^2 - 72x + 9, [-4, 5] absolute minimum value (4, -199) absolute maximum value (-3, 144)