Check homework: Concavity and the Second Derivative

[chettacheez]

Hey guys

I have a chance to do this extra credit homework in hope of raising my final grade in the class. It deals with concavity and the second derivative. This was a while ago and it just so happens my notes are 300 miles away. Would you guys check over my answers? I am mostly concerned with number 1 and 2.


Thanks!

 

[DrPhil]

#1 looks good to me. I would have made a cusp at x=4 . continuous function but discontinuous derivative. But what you did (vertical tangent at x=4) is more elegant.

#2 a) There is only a relative max at x=1. Even though the derivative is 0 at x=+1, the sign of f'(x) does not change sign.
b) yes, inflections at x=0 and at x=1 - state why (2nd derivative zero)
c) The left side of your plot should look like an inverted (concave down) parabola with axis at x=-1. Ar x=0 the curvature changes to concave up, and at x=1 the curve is horizontal, and the curvature returns to concave down.

#3 correct (justification?)

#4 correct (justification?)

#5 c) you may have forgotten that the plot is the derivative, not f
d) ditto. BTW, notice that the derivative is not defined at x=0 or at x=10. Just on the open interval (0,10).

On a real test, you MUST write down the justifications!

Good luck!

 

[chettacheez]

Thank you so much for the feedback. Good call on the justifications too, I completely forgot about them!