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.