Question

"Find all relative extrema. Use the second derivative test where applicable" f(x) = x^3 - 3x^2 + 3....how do I know if I am supposed to take the first derivative or the second derivative?

Answer 1

toot toot hhahaahaha

Answer 2

can somebody plez help me? :)

Answer 3

Always take the first derivative first:
\[f(x)=x^3-3x^2+3~~\Rightarrow~~f'(x)=3x^2-6x\]
Find the critical points; set \(f'(x)=0\) and solve for \(x\).

These values are points on the curve \(f(x)\) where the slope of the tangent line is horizontal. However, this doesn't guarantee that these points are extrema.

Examples:
To determine if a critical point is an extremum, you can take the second derivative to determine the concavity of the curve to the left and right of any critical point. Notice how the minimum occurs when the curve is concave up to both sides of the critical point; the max occurs when the curve is concave down; and no extremum occurs when the curve changes concavity to either side.
\[f'(x)=3x^2-6x~~\Rightarrow~~f''(x)=6x-6\]