Differentiation

• At a stationary point, the gradient of the curve is Zero
• If a stationary point is a local maximum point, the gradient of the curve is positive just to the left of the stationary point and negative just to the right of the stationary point
• If a stationary point is a local minimum point, the gradient of the curve is negative just to the left of the stationary point and positive just to the right of the stationary point
• If the second derivative is positive at a stationary point, the stationary point is a local minimum
• If the second derivative is negative at a stationary point, the stationary point is a local maximum.
• What does the second derivative tell you about a curve


  The second derivative is the rate of change of the first derivative, so its value tells you about the rate of change of the gradient of the curve.
• If the second derivative is negative at a stationary point, this means that the stationary point is a local maximum.

   

  Explain why this is the case.

  If the second derivative is negative, this means that the gradient is decreasing across the stationary point.

   

  The gradient is zero at the stationary point, so the gradient must be changing from positive to negative and so the stationary point is a local maximum.
• If the first and second derivatives are both zero at a point on a curve, does this mean that the point must be a stationary point of inflection?

  If the first and second derivatives are both zero at a point on a curve, the point is a stationary point, but it could be either a stationary point of inflection, a local maximum or a local minimum.

   

  The questions in the next activity will help you to understand this.