Is f(x)=x^4-4x^3+x-4 concave or convex at x=-1?

1 Answer
Mar 22, 2016

convex

Explanation:

To find that out, we need to get the second derivative first.

Getting the first derivative.

[1]" "f'(x)=d/dx(x^4-4x^3+x-4)

We can easily find this using power rule.

[2]" "f'(x)=4x^3-12x^2+1

Getting the second derivative.

[1]" "f''(x)=d/dx(4x^3-12x^2+1)

Use power rule again.

[2]" "f''(x)=12x^2-24x

Now that we know the second derivative, we will evaluate f''(x) at x=-1 to check its concavity.

• If f''(x)>0, then it is concave up or convex
• If f''(x)<0, then it is concave down or concave

[1]" "f''(-1)=12(-1)^2-24(-1)

[2]" "f''(-1)=12+24

[3]" "f''(-1)=36

Since f(x) is 36 at x=-1 and 36 is greater than 0, then f(x) is convex at x=-1.