Find the inflection point, if any, for the function f(x) = x^3 + 3x^2 + 1

Answer 1

f(x)=x^3+3x^2+1

f'(x)=3x^2+6x

f''(x)=6x+6

Set the second derivative to 0 to check for any inflection points, solve for x.

6x+6=0; 6x=-6; x=-1

Now test some points around -1 to see concavity. f''(0)=6 - concave up, and f''(-10)=-54 - concave down. Since points on either side of the point are different concavities, then this must be an inflection point at x=-1.

Answer 2

f'(x)=3x²+6x

f"(x)=6x+6 and f"(x)=0 when x = -1

The point of inflection occurs when x=-1.