Derivative Tests: Read the Shape of a Graph

This section-sized Calculus I lesson shows how the signs of the first and second derivatives describe a graph’s shape. You will build sign charts, state increasing and decreasing intervals, classify local extrema, verify inflection points, and apply the Second Derivative Test only when its conditions support a conclusion. These sign-based tests and their conditions follow standard Calculus I curve-analysis methods. 1

Two complete examples develop the full exam-credit workflow: f(x)=x³−3x²−9x+1 and g(x)=x⁴−4x². A final diagnostic compares x⁴, −x⁴, and x³ at zero to show why f″(c)=0 is inconclusive rather than a maximum, minimum, or inflection-point verdict.