7. (13pt) Find the intervals on which the function f(x) =...

7. (13pt) Find the intervals on which the function $f(x) = 2x^3 + 3x^2 - 12x$ is increasing and those on which it is decreasing. Sketch neatly the graph of $y = f(x)$ and label the local maxima and minima.

Transcript

00:01 All right, to find out where a function is increasing and decreasing, we need to find the derivative so that we can find the critical values.

00:10 And then once we find the critical values, we'll test to see whether the derivative is positive or negative.

00:17 All right.

00:17 So this are products.

00:18 We're going to use the product rule.

00:20 It's the first times the derivative of the second, two -thirds x minus one to the minus one third, times the derivative of x minus one, which is one.

00:32 First derivative the second plus the second times the derivative of the first.

00:39 Okay, so that is 2x over 3 times x minus 1 to the 1 3 plus x minus 1 to the 2 thirds over 1.

00:52 So i'm going to multiply by 3x minus 1 to the 1 3 3 to the 1 3rd here to get a common denominator.

01:03 So now i have 2x plus.

01:06 Well, when i multiply x minus 1 to the 2 thirds times x.

01:10 Minus 1 to the 1 third, i get x minus 1.

01:13 So i get 3 x minus 1 there over 3x minus 1 to the 1 3.

01:22 So 2x plus 3x minus 3 over 3x minus 1 to the 1 3.

01:34 So 5x minus 3 over 3 times x minus 1 to the 1 3 is the derivative.

01:44 Okay, f prime of x equals 0.

01:48 When 5x minus 3 over 3x minus 1 to the 1 3 3 3 .3 equals 0 over 1.

02:00 Cross multiply, 5x minus 3 equals 0 or x is 3 5ths.

02:08 But there's also another critical point.

02:10 F prime of x is undefined or does not exist when 3x minus 1 to the 1 3 .3.

02:23 So if the denominator is zero, you also get a critical value.

02:30 Divide by three, cube both sides, x equals one.

02:34 So we have critical values at three -fifths and one.

02:40 So now draw a number line.

02:43 And now we're gonna check the derivative, 5x minus three over three x minus one to the one third.

02:51 Pick a number bigger than one, any number that you like, 100, a million, 1 .5.

02:58 Let's just try to 5 times 2 minus 3.

03:03 That is positive over 3 times 5 minus 1 to the 1 3rd.

03:09 That is positive.

03:10 Positive divide by positive is positive.

03:13 So that's increasing.

03:18 All right...

Question

7. (13pt) Find the intervals on which the function $f(x) = 2x^3 + 3x^2 - 12x$ is increasing and those on which it is decreasing. Sketch neatly the graph of $y = f(x)$ and label the local maxima and minima.

Please give Ace some feedback