How do I find the absolute minimum and maximum of a function using its derivatives?

Question

Aurora Alvarado · Accepted Answer

See below

Derivating a function will give us its gradient for every #(x,y)# on it. At the minima and maxima, the gradient of the function will be zero. So once we've found the derivative, if we want to find the minima and maxima, we set the derivative equal to zero and solve for #x#. Once we've found the value of #x#, we should calculate #f''(x)#, and this will tell us if the stationary point at #x# is a minimum or maximum. We can then use a table of values to see if #y# is increasing or decreasing around the turning point and this will help identify if it's a local or absolute stationary point.

Isabella Ackerman · Answer

undefined To find the absolute minimum and maximum of a function using its derivatives, follow these steps:

1. Find the critical points of the function by setting its derivative equal to zero and solving for $ x $.
2. Determine the intervals between the critical points.
3. Evaluate the function at the critical points and at the endpoints of the intervals.
4. The lowest function value among these values is the absolute minimum, and the highest function value is the absolute maximum.