Stationary Points and Turning Points

Stationary Points and Turning Points

Differentiation and Stationary Points

  • Differentiation is a vital tool in calculus that involves determining an object’s rate of change. The derivative of a function can help in finding its stationary points.

  • Stationary points are where the derivative (slope) of the function is zero, meaning the curve ‘stops moving’ either in the positive or negative direction. Mathematically, a point x = a is a stationary point if f’(a) = 0.

  • Stationary points can be of three types: local minima, local maxima and points of inflection.

  • A local minimum is a point where the function takes the smallest value within a certain interval.

  • A local maximum is a point where the function takes its highest value within a certain interval.

  • At a point of inflection the function changes concavity – it switches between being concave up (shaped like a U) to concave down (shaped like an n), or vice versa.

Determining the Nature of Stationary Points

  • To determine whether a stationary point is a minimum, maximum or inflection point, we can apply the second derivative test.

  • If you differentiate the function a second time to find the second derivative, f’‘(x), and substitute the x-coordinate of the stationary point:

    • if f’‘(x) is positive, the point is a local minimum,
    • if f’‘(x) is negative, the point is a local maximum,
    • and if f’‘(x) is zero, the test is inconclusive.

Turning Points

  • A turning point is a point where the graph of a function “turns around.” Hence, all turning points are stationary points, but not all stationary points are turning points.

  • A function can have many stationary points, but only those points where the function changes direction are called turning points.

  • For a turning point, the tangent to the curve is horizontal. That is why the gradient of the curve (or the derivative of the function) is zero at a turning point.

Remember, a good understanding and application of differentiation and the second derivative test are fundamental in solving problems related to stationary and turning points.