Show that 5 is a critical number of the function g(x)=2+(x-5)^3 but g does not have a local extr

step 1 according to the given question Gx is 2 plus x minus 5 raised to power 3 we are need to find if

Show that 5 is a critical number of the function...

Key Concepts

Critical Number

A critical number of a function is a point in its domain where either the derivative is zero or the derivative does not exist. Identifying critical numbers is important because they are potential candidates where the function might have local maximum or minimum values.

Derivative

The derivative of a function provides information about the rate of change of the function with respect to its variable. In the context of optimization, computing the derivative and checking where it is zero or undefined is fundamental in locating critical points.

Local Extreme Value

Local extreme values, which include local minima and maxima, are points in the function's domain where the value of the function is lower or higher than nearby points. Even though critical numbers often indicate potential extreme values, additional analysis is required to confirm whether such a point is indeed a local extremum.

First Derivative Test

The first derivative test is a method used to determine whether a critical number corresponds to a local extreme value. By examining the sign of the derivative on either side of the critical number, one can decide if the function changes from increasing to decreasing (indicating a local maximum) or from decreasing to increasing (indicating a local minimum). If the derivative does not change sign, the point is not a local extremum.

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