Introduction

In mathematics, maxima and minima (the plural of maximum and minimum) are points of a function at which the function takes its largest value (maximum) or smallest value (minimum), respectively. In the case of a local maximum or minimum, this value is also the largest value or smallest value in a neighboring interval. Maxima and minima are therefore local extrema of functions.

What is a function?

A function is a mathematical relation between two sets, usually denoted by an equation. The first set is called the domain and the second set is called the range. The function assigns a unique output to every input. In other words, for every element in the domain, there is a corresponding element in the range.

What is a maxima?

In mathematics, the maxima (the plural of maximum) are the highest points of a function. The maxima of a function can be found by taking the derivative and setting it equal to zero. The points where the derivative is zero are called the critical points. The critical points can be maxima, minima, or inflection points. To determine which type ofcritical point it is, you must take the second derivative and evaluating it at the critical point. If the second derivative is positive, it is a maximum; if negative, it is a minimum; and if zero, it is an inflection point.

How to find the maxima for a function

To find the maxima for a function, you need to take the derivative of the function and set it equal to zero. The maxima will be the points where the derivative is equal to zero.

Examples

To find the maxima for the function

-Take the derivative of the function.
-Set the derivative equal to zero and solve for x.
-Plug in your x value into the original function to see if it is a maximum or minimum point. If it is a maximum point, it will be positive and if it is a minimum point, it will be negative.

For example, let’s say you have the function f(x)=-2x^2+5x+3 and you want to find its maxima.

To do this, you would take the derivative of the function which would give you f'(x)=-4x+5. Then, you would set this equal to zero and solve for x which would give you x=1/4. You would then plug this value into the original function to get f(1/4)=11/8 which is a maximum point.

Conclusion

To find the maxima for the function, we take the derivative and set it equal to zero. This gives us the equation:

0 = -2x^3 + 12x^2 – 20x + 8

We can use the Quadratic Formula to solve this equation and find that the maxima for the function is at x = 2.