Question

A function is given. (a) Find all the local maximum and minimum values of the function and the value ofat which each occurs. State each answer rounded to two decimal places. (b) Find the intervals on which the function is increasing and on which the function is decreasing. State each answer rounded to two decimal places.

$U\left(x\right)=x\sqrt{6-x}$

Short answer

1. The local maximum value is at $x=4$and local maximum value is $5.657$
2. The function$U$ is increasing in $(-\infty ,4)$and decreasing in$(4,6)$

Step-by-step solution

Step 1. Concept of local maxima and minima.

The function value$U\left(a\right)$ is a local maximum value of$U$ if$U\left(a\right)\ge U\left(x\right)$

And, The function value $U\left(a\right)$is local minimum value of$U$ if$U\left(a\right)\le U\left(x\right)$

Step 2. Graph of the function U.

$U\left(x\right)=x\sqrt{6-x}$

Step 3. Determining the local maximum and local minimum of the function U.

$U\left(x\right)=x\sqrt{6-x}$

At$x=4$, the function has local maximum point

There is no local minimum point

The value of at$x=4$ is

$$\begin{gathered} U=4\sqrt{6-4} \\ U=4\sqrt{2} \\ U=5.657 \end{gathered}$$

Step 4. Concept of increasing and decreasing function.

$U$is increasing on an interval I if$U\left(x_1\right)<U\left(x_2\right)$ whenever$x_1<x_2$ in I

$U$is decreasing on an interval I if$U\left(x_1\right)>U\left(x_2\right)$ whenever$x_1<x_2$ in I

Step 5. Determining the domain and range of the function.

The domain of the function

$U\left(x\right)=x\sqrt{6-x}$Is$x\ge 6$

That is$x\in \left[6,\infty \right)$

And Range is$\left(-\infty ,5.657\right]$

Step 6. Determining the interval in which the function is increasing or decreasing.

The function $U\left(x\right)=x\sqrt{6-x}$is increasing in the interval $(-\infty ,4)$and the function is decreasing in the interval $(4,6)$.