Concave & Convex Functions (AQA A Level Maths): Revision Note

Applications of the Second Derivative

What applications of the second derivative do I need to know?

• The second order derivative (or simply second derivative) is encountered at AS level

  • At AS level second derivatives are used to help determine the nature of a stationary point

• At A level you need to be able to use the second derivative to determine if a function is convex or concave on a given interval

  • Where a function is convex its graph 'curves up' and its tangent lines lie below the graph

  • Where a function is concave its graph 'curves down' and its tangent lines lie above the graph

• It's the sign of the second derivative (+ or -) that's important:

  • A function f(x) is convex on a given interval if and only if f''(x) ≥ 0 for every value of x in the interval

  • A function f(x) is concave on a given interval if and only if f''(x) ≤ 0 for every value of x in the interval