For an increasing function f(x) in the interval (a, b), f’(x) > 0 in the interval a ≤ x ≤ b

For a decreasing function f(x) in the interval (a, b), f’(x) < 0 in the interval a ≤ x ≤ b

The points where f(x) stops increasing and begins to decrease are called maximum points

The points where f(x) stops decreasing and begins to increase are called minimum points

A point of inflection is a point where the gradient is at a maximum or minimum value in the neighbourhood of the point

A stationary point is a point of zero gradient. It may be a maximum, a minimum or a point of inflection

To find the coordinates of a stationary point:

Find  (The gradient function)

Solve the equation f’(x) = 0 to find the value, or values, of x

Substitute into y = f(x) to find the corresponding values of y

The stationary value of a function is the value of y at the stationary point. You can sometimes use this to find the range of a function

You may determine the nature of a stationary point by using the second derivative

In problems where you need to find the maximum or minimum value of a variable y, first establish a formula for y in terms of x, then differentiate and put the derived function equal to zero to then find x and then y