Business Mathematics and Statistics · Ch 6 — Applications of Differentiation (incl. Business/Economics applications, Maxima/Minima, Partial Derivatives)
Maxima and Minima of a Function — First and Second Derivative Tests
Maxima and Minima of a Function — First and Second Derivative Tests
Many business questions boil down to finding the single best value of something — the output that maximises profit, the price that minimises average cost. For a differentiable function , this is done using its derivatives.
Step 1 — find the stationary (critical) points. A stationary point is a value of where the tangent to the curve is horizontal, i.e. . Solve this equation for every real root; each root is a candidate for a local maximum or minimum (the function's slope momentarily stops rising or falling there).
Step 2 — classify each stationary point. Two equivalent tests are used:
First Derivative Test
Check the sign of just before and just after the stationary point . If changes from positive to negative, has a local maximum at . If it changes from negative to positive, has a local minimum at . If the sign does not change, is neither (an inflection-type stationary point).
Second Derivative Test
Compute at the stationary point. If , has a local maximum at . If , has a local minimum at . If , the test is inconclusive and the first derivative test must be used instead. …
A value of at which ; a candidate location for a local maximum or …
A stationary point at which changes from positive to negative (equivalently ) — the function's value there is highe …
A stationary point at which changes from negative to positive (equivalently ) — the function's value there is lowe …