How to Analyze a Function Using First and Second Derivatives

The first derivative describes increasing and decreasing behavior. The second derivative describes how the slope changes, which determines concavity. Together they provide a detailed picture of a function’s shape.

What the first derivative tells you

• f′(x)>0: the function is increasing.
• f′(x)<0: the function is decreasing.
• A sign change at a critical point can identify a local extremum.

What the second derivative tells you

• f″(x)>0: the graph is concave up.
• f″(x)<0: the graph is concave down.
• A change in concavity at a point on the graph identifies an inflection point.

Worked example: f(x)=x^3-3x

f′(x)=3x^2-3
f″(x)=6x

The first derivative is zero at x=-1 and x=1. The function increases on (-∞,-1) and (1,∞), and decreases on (-1,1).

The second derivative is zero at x=0. It is negative for x<0 and positive for x>0, so the graph changes from concave down to concave up. Since f(0)=0, the inflection point is (0,0).

Combine the information

Inflection points require a sign change

Solving f″(x)=0 only produces candidates. Confirm that concavity actually changes and that the point belongs to the graph.

Create a combined analysis table

Open the Variation Table tool and select First and Second Derivative Analysis Table. The format combines signs of f′ and f″ with monotonicity, concavity, extrema, and inflection information.

Check the graph

Use the Graphing Calculator to compare the symbolic analysis with the visible curve. The graph should rise and fall according to f′, and bend according to f″.

Common mistakes

• Using f″=0 alone to claim an inflection point.
• Confusing concave up with increasing.
• Forgetting domain breaks.
• Evaluating critical values in the derivative instead of the original function.
• Reading approximate graph coordinates as exact values.

Build the combined derivative-analysis table →

For visual regions rather than derivative behavior, continue with graphing and shading inequalities.