Is f(x) = 8x^4+3x^3 + 4x^2+x-4 concave or convex at x=11?

1 Answer
Feb 1, 2018

Concave

Explanation:

The concavity of a function is given by it's third derivative:
When f''(x)>0, the function is concave and when f''(x)<0, it's convex.

Let's determine the first derivative:

f'(x)=(8x^4+3x^3+4x^2+x-4)'

Given (u+v)'=u'+v', (ku)'=ktimesu' and (x^n)'=ntimesx^(n-1:

f'(x)=32x^3+9x^2+2x+1

Now, determine the second derivative:

f''(x)=(32x^3+9x^2+2x+1)'

f''(x)=96x^2+18x+2

Plug in x=11:

f''(11)=96times11^2+18times11+2=11.816

As f''(11)>0, the function is concave at x=11