Question

Show that 5 is a critical number of the function \(g(x) = 2 + {(x - 5)^3}\) but \(g\)does not have a local extreme value at \(5\).

Short answer

\(5\) Is the critical number of the function \(g(x) = 2 + {(x - 5)^3}\) but the curve of \(g(x)\) does not have a local extreme value at \(5\).

Step-by-step solution

Given function

The given function is \(g(x) = \left( {2 + {{(x - 5)}^3}} \right)\).

Definition of Critical Number

A critical number of a function\(f\)is a number\(c\), if it satisfies either of the below conditions:

(1)\({f^\prime }(c) = 0\)

(2)\({f^\prime }(c)\)Does not exist.

Check the conditions of Critical Number

Find the first derivative of\(g(x)\).

\(\begin{aligned}{c}{g^\prime }(x) = \frac{d}{{dx}}\left( {2 + {{(x - 5)}^3}} \right)\\{g^\prime }(x) = 3{(x - 5)^2}\end{aligned}\) ...... (1)

Substitute\(x = 5\)in equation (1).

\(\begin{aligned}{c}{g^\prime }(5) = 3{(5 - 5)^2}\\{g^\prime }(5) = 0\end{aligned}\)

Hence, it satisfies the condition of the definition of critical number.

Therefore, it is proved that\(5\)is a critical number of the function\(g(x) = 2 + {(x - 5)^3}\).

Also observe that the derivative of\(g(x)\)will be positive for any value of\(x\).

So, the graph of\(g(x)\)must be an increasing graph.

From this observation, it can be concluded that the function has no local extreme as the increasing graph does not have local extreme.

Therefore,\(g(x)\)does not have any local extreme at\(5\).

Hence, \(5\) is the critical number of the function \(g(x) = 2 + {(x - 5)^3}\) but the curve of \(g(x)\) does not have a local extreme value at \(5\).