# For what values of x is #f(x)= 2x^3-9x # concave or convex?

##### 1 Answer

I try not to confuse myself over "concave" vs. "convex". Instead I think about it as concave up or concave down.

The **first** derivative equals

It is the **second** derivative at each of these points that tells you which of these three they are. Positive, if concave up, and negative, if concave down.

#f'(x) = 6x^2 - 9#

(power rule;

#f''(x) = 12x#

For us to find where the extrema are:

#0 = 6x^2 - 9#

#=> x^2 = 9/6 = 3/2#

#=> x = pmsqrt(3/2)#

And to find which one is concave up/down or an inflection point, we take

Thus, **up** (a minimum) at **down** (a maximum) at

Indeed,

graph{2x^3 - 9x [-10,10, -10.14, 10.13]}