Find all local extreme values of f ( x ) = (2 /3) x ^3 + 7 x^ 2 + 20 x − 100 .

Question

I find as the first local extreme value from left to right is the maximum at (-2, -352/3) which is wrong.

I found the derivative of the function to be 2x^2+14x+20 and set it = to 0 and solve to get x=-5,-2.

Then I find the second derivative and plug in the x values I got for the first derivative into the second one and get f''(-2)=6 and f''(-5)=-6. Then I plug in -2 and -5 into the original equation as f(-2) and (-5) and get f(-2)=-352/2 and f(-5)=-325/3. The points I get as a result are (-2, -352/3) and (-5, -325/3) which are both wrong. Did I make a math error or leave out a step?? Please help!

Answer 1

What you missed is that -5 is to the left of -2.

Answer 2

Hello Bri S.,

Thanks for showing your work so clearly. You did a lot of good work. I think you made two errors right at the end when you substituted x = -2 and x = -5 into the original function. Check your computations for both points.

Since x = -2 is concave up, it should be a local minimum and x = -5 is concave down so it should be a local maximum once you get the points. In other words, everything looks right except for the y-value of your local extreme points.

Tom S.