SL 5.6—Stationary points, local max and min

Stationary Point: A point on a curve where the derivative equals zero, i.e. $f^{\prime }(x)=0$. At these points, the tangent to the curve is horizontal and the function is momentarily "not changing".

Stationary points are important because they tell us where a function stops increasing and starts decreasing (or vice versa). There are two main types we focus on at SL:

• Local maximum: The function reaches its highest value in a small neighbourhood around that point. The curve rises to that point, then falls away.
• Local minimum: The function reaches its lowest value in a small neighbourhood around that point. The curve falls to that point, then rises away.
• Horizontal point of inflection: The derivative equals zero, but the function does not change direction , it continues increasing (or decreasing) on both sides. This is a stationary point that is neither a local max nor a local min. You should be aware this can occur, though it is less commonly tested at SL.

Recall that the derivative $f^{\prime }(x)$ gives the gradient (slope) of the tangent to the curve at any point $x$.

• When a function is increasing, its gradient is positive: $f^{\prime }(x)>0$
• When a function is decreasing, its gradient is negative: $f^{\prime }(x)<0$
• At the exact moment a smooth, differentiable function switches from increasing to decreasing (or vice versa), the gradient passes through zero: $f^{\prime }(x)=0$

This is why stationary points are found by solving $f^{\prime }(x)=0$. At a local maximum, the function transitions from increasing to decreasing. At a local minimum, it transitions from decreasing to increasing.