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}