# Consider the function f(x) = 9 - |x - 3 | a. Find the critical numbers of the function. b....

## Question:

Consider the function

{eq}f(x) = 9 - |x - 3 | {/eq}

a. Find the critical numbers of the function.

b. Find the open intervals where the function is increasing or decreasing.

c. Apply the First Derivative Test to identify the relative extremum.

## Qualitative Features of Graphs

To see where a function {eq}\displaystyle y=f(x) {/eq} is increasing or decreasing, we look at the sign of its first derivative.

{eq}\displaystyle \text{ if } f'(x)>0 \text{ then } f(x) \text{ is increasing} \displaystyle \text{ and where } f'(x)<0 , f(x) \text{ is decreasing}. {/eq}

Critical points 'are the points 'c from the given domain of the function

where {eq}\displaystyle f'(c)= 0, \text{ or } f'(c) - \text{ does not exist}. {/eq}

If the derivative changes from negative to positive at a critical point, the point is a local minimum.

a. To find the critical points of the function {eq}\displaystyle f(x)=9-|x-3| {/eq}

we will first use the definition of the absolute value to rewrite the function as

{eq}\displaystyle f(x)=9-|x-3|=\begin{cases} 12-x, x\geq 3\\ 6+x, x<3 \end{cases} {/eq}

The critical points are given by the first derivative being zero or undefined.

Because the first derivative is {eq}\displaystyle f'(x)=\begin{cases} -1, x\geq 3\\ 1, x<3 \end{cases} {/eq}

is not defined at {eq}\displaystyle x=3, {/eq} \because the side limits are not equal {eq}\displaystyle \lim_{x\to 3^-} f'(x)=1\neq \lim_{x\to 3^+} f'(x)=-1. {/eq}

therefore {eq}\displaystyle \boxed{ \text{ the critical point is at }x=3}. {/eq}

b. The function is {eq}\displaystyle \boxed{\text{ increasing on } (-\infty, 3) \text{ and it is decreasing on } (3,\infty)}. {/eq}

because at the left of 3, the line has positive slope, while at the right of 3, the line has negative slope.

c. The derivative function, {eq}\displaystyle f'(x)=\begin{cases} -1, x\geq 3\\ 1, x<3 \end{cases} {/eq}

changes from positive to negative at the critical point {eq}\displaystyle x=3, {/eq} therefore {eq}\displaystyle \boxed{x=3 \text{ is a relative maximum } }. {/eq}