# Using Calculus to Find Maxima and Minima in Simple Problems

## Using Calculus to Find Maxima and Minima in Simple Problems

# Understanding the Concept

- Calculus is a powerful tool that allows us to determine the maximum or minimum values of certain quantities.
- These
**maxima**or**minima**may represent the highest or lowest points on a graph, the maximum or minimum values of a function, etc. - The procedure is applicable to a wide range of simple to complex problems.

# Differentiation Process

- Differentiation is a Calculus technique that helps us find the rate at which a quantity is changing at a particular point.
- In the context of maxima and minima problems,
**differentiation**helps us find the point on a graph where the slope is zero. - A function reaches its maximum or minimum when the derivative of the function (also known as its
**gradient**or slope) is zero.

# Steps to Solve Problems

- Initially, express the problem as a function, f(x).
- Next, find the derivative f’(x) of the function.
- Set f’(x) to zero and solve for x to find potential
**critical points**. - Use the second derivative test to determine whether each critical point is a maximum or minimum. If the second derivative, f’‘(x), is positive at the critical point, the function shows a minimum at that point; whereas if f’‘(x) is negative, it indicates a maximum.

# Practicing Example

- Consider a function f(x) = 12 - 3x^2. Its derivative is f’(x) = -6x. Setting f’(x) to zero, we get -6x = 0, or x = 0 as the critical point.
- The second derivative f’‘(x) = -6. Since this is negative, it indicates that the point x = 0 is a maximum point. Following the technique and practising more problems would improve problem-solving skills in calculus.

# General Tips

- While applying this concept to real-world problems, make sure to clearly define what you’re trying to maximise or minimise.
- Keep practising multiple problems to get comfortable with the concept and the process involved.

Remember, mastery of calculus concepts requires practise and patience!