# Use of Differentiation to Find Maxima and Minima Points on a Curve

## Use of Differentiation to Find Maxima and Minima Points on a Curve

# Understanding Maxima and Minima Points

## Concepts and Definitions

**Maxima**and**Minima**are important aspects of differentiation used to define the highest and lowest points on a curve respectively.- A
**maximum point**on a curve refers to a point where the curve changes from an increasing to a decreasing gradient. - A
**minimum point**on a curve refers to a point where the curve changes from a decreasing to an increasing gradient.

## Process of Finding Maxima and Minima

- To find maxima and minima points, first find the derivative of the function. The derivative will provide the gradient function.
- Set the derivative equal to zero and solve for
*x*. These*x*values are critical and could be the*x*values of points of maxima or minima. - To verify whether the points are maximum or minimum, use the second derivative test, if possible.

## The Second Derivative Test

- If the second derivative of a function at a point is negative, then the function reaches a
**maximum**at that point. - If the second derivative is positive, the function has a
**minimum**at that point. - If the second derivative is zero, the test is inconclusive. Consider using the first derivative test.

## Applications of Maxima and Minima

- Maxima and minima have practical applications in various fields such as economics, physics, engineering, and more.
- They help in understanding the points of maximum profit, minimum cost, maximum efficiency, etc.

## Additional Points

- Be aware of the fact that maximum and minimum points can also occur on the endpoints of a function’s domain.
- Not all turning points will be maximum or minimum. Points where a function changes direction but doesn’t achieve a local maximum or minimum are called
**points of inflexion**.

Remember, practise is crucial in mastering the method for finding maxima and minima using differentiation. Recurring practise of several types of questions will build confidence in this area.