Local maxima and minima calculator

Local Maxima and Minima Calculator

A local Maxima and Minima calculator is an online tool used to find the local maxima and local minima of math functions. Extrema are the lowest or highest points of a function. They come in two principal types:

1. Absolute extrema which includes

Global Maximum:  The highest point of the function.

Global Minimum:  The lowest function of the function.

2. Local extrema which includes

Local Maximum: At x = c if f(c) is greater than the function values, then that point is a local maximum. Formally, for all x that is close to c, f(c) ≥ f(x).

Local Minimum: If f(c) is less than the function values at a point x = c, then that point is a local minimum. In formal terms, f(c) ≤ f(x) for all x that are close to c.

How to use our Local Maxima and Minima Calculator:

Follow this procedure to use our calculator:

Step 1: Input the Function: 

Enter the mathematical function for which you want to find the local maxima and minima in a given input field.

Step 2: Calculate: 

Click the "Submit" button to compute the outcome.

How to compute local maxima and minima?

To figure out how to compute the local maxima and minima, follow the below example:

Example 1:

Let's take a function f(x) = 2x^3 - 3x^2 - 12x + 5 as an example

Calculation steps:

Take its first derivative:

 f’(x) = 6x^2 - 6x - 12

Now to find critical points put f ‘(x) equal to zero

6x^2 - 6x - 12 = 0

6(x^2 - x - 2) = 0

6(x-1)(x-2) = 0

x = -1,  x = 2           -> critical points

Now take the second derivative of f’(x)

f’’(x) = 12x - 6

As we got x = 2, -1 as critical points, so we first evaluate f’’(x) at x = -1

f’’(2) = 12(-1) - 6 = -18

As f′′(−1) < 0 , x = -1 is a local maximum

Now we evaluate f’’(x) at x= 2

f′′(2) = 12(2) − 6 = 24 − 6 = 18

As f′′(2) > 0, x = 2 is a local minimum.

Now let's evaluate Function Values at Critical Points:

At x = 2 

f(2) = 2( 2)^3 - 3(2)^2 - 12(2)+5 = -15

At x = -1

f(-1) = 2(-1)^3 - 3(-1)^2 - 12(-1) + 5 = 12

So when f(x) = 12 the local maximum is -1, when f(x) = -15 the local minimum is 2. 

Faqs

Which functions is the calculator capable of processing?

The calculator can calculate many functions including exponential, logarithmic, and trigonometric functions.

How does the calculator manage discontinuities or undefined points within a function?

The calculator detects and manages discontinuities or undefined points in a function by examining the function's domain. It produces outputs for the continuous segments of the function while highlighting locations where the function is not defined.