# Is #f(x)=-5x^5-2x^4-2x^3+14x-17# concave or convex at #x=0#?

##### 1 Answer

Neither. It is a point of inflection.

#### Explanation:

Convexity and concavity are determined by the sign of the second derivative.

- If
#f''(0)>0# , then#f(x)# is convex when#x=0# . - If
#f''(0)<0# , then#f(x)# is concave when#x=0# .

Find the function's second derivative.

#f(x)=-5x^5-2x^4-2x^3+14x-17#

#f'(x)=-25x^4-8x^3-6x^2+14#

#f''(x)=-100x^3-24x^2-12x#

Find the sign of the second derivative at

#f''(0)=0#

Notice that the sign of the second derivative is neither positive nor negative. This means that the function is neither convex nor concave. This means that is may be a point of inflection.

We can check a graph of the function:

graph{-5x^5-2x^4-2x^3+14x-17 [-2.5, 2.5, -120, 100]}

Graphically,