# Is #f(x)=(x-3)^3-x+15# concave or convex at #x=3#?

##### 1 Answer

Neither.

#### Explanation:

You have to find the second derivative to answer this question.

I would use the chain rule to differentiate

#f'(x) = 3(x- 3)^2 - 1#

Now it's easy enough to expand to find the second derivative using the power rule.

#f'(x) = 3(x^2 - 6x - 9) - 1#

#f'(x) = 3x^2 - 18x - 27 - 1#

#f'(x) = 3x^2 - 18x - 28#

Now find the second derivative.

#f''(x) = 6x - 18#

We now test the sign of the second derivative at

#•f''(x) > 0# , at#x = a# then#f(x)# is concave at#x = a#

#•f''(x) < 0# , at#x = a# then#f(x)# is convex at#x = a#

We have:

#f''(3) = 6(3) - 18 = 18 - 18 = 0#

So

Hopefully this helps!