Finding and classifying turning points

At a stationary point the tangent is horizontal, so \(\dfrac{dy}{dx}=0\). Solve this equation for x, then substitute into the original function to find y. A maximum has gradient changing from positive to negative; a minimum changes from negative to positive.

Example

For \(y=x^3-3x\), \(\dfrac{dy}{dx}=3x^2-3=0\) gives \(x=-1,1\). The points are \((-1,2)\) and \((1,-2)\).

Thus \((-1,2)\) is a local maximum and \((1,-2)\) a local minimum. Alternatively differentiate again: a positive second derivative at a stationary point indicates a minimum; a negative one indicates a maximum. If it is zero, that test does not decide. Do not label a point a maximum or minimum merely because its first derivative is zero.