Is #f(x)=3x^5-x^3+8x^2-x+3# concave or convex at #x=-4#?

1 Answer
Oct 3, 2017

Concave down

Explanation:

To find the concavity of a function #f(x)#, you will need to determine the second derivative #f''(x)#, and then evaluate at at the given point. If the result of #f''(a) > 0#, the function is concave up; if the result is negative, it it concave down.

#f(x) = 3x^5 - x^3 + 8x^2 - x + 3#

#f'(x) = 15x^4 - 3x^2 + 16x = 1#

#f''(x) = 60x^3 - 6x + 16#

#f''(-4) = 60(-4)^3 - 6(-4) + 16 = -3,800 < 0#

Thus, #f(x)# is concave down at #x = -4#

graph{3x^5 - x^3 + 8x^2 - x + 3 [-4.59, 2.72, -5000.82, 500.84]}