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