Question

consider the following function. use a graphing utility to confirm your answers for parts (a) through (c). (if an answer does not exist, enter dne.)

$f(x)=x^{2}-10x$

(a) find the critical numbers of f. (enter your answers as a comma-separated list.)

$x=$

(b) find the open intervals on which the function is increasing or decreasing. (enter your answers using interval notation.)

increasing

decreasing

(c) apply the first derivative test to identify all relative extrema.

relative maximum $(x,y)=(quad)$

relative minimum $(x,y)=(quad)$

Explanation:

Step1: Find the derivative

The function is \(f(x)=x^{2}-10x\). Using the power rule \((x^n)^\prime = nx^{n - 1}\), the derivative \(f^\prime(x)=2x-10\).

Step2: Find critical numbers

Set \(f^\prime(x) = 0\). So, \(2x-10=0\). Solving for \(x\):

$$
LATEXBLOCK0
$$

Step3: Determine intervals of increase and decrease

• For the interval \((-\infty,5)\), let's test \(x = 0\). Then \(f^\prime(0)=2\times0 - 10=- 10<0\). So the function is decreasing on \((-\infty,5)\).
• For the interval \((5,\infty)\), let's test \(x = 6\). Then \(f^\prime(6)=2\times6 - 10 = 2>0\). So the function is increasing on \((5,\infty)\).

Step4: Apply the First - Derivative Test

Since \(f^\prime(x)\) changes sign from negative to positive at \(x = 5\).

• There is no relative maximum (because the function changes from decreasing to increasing).
• To find the \(y\) - value of the relative minimum, substitute \(x = 5\) into \(f(x)\): \(f(5)=5^{2}-10\times5=25 - 50=-25\)

Answer:

(a) \(x = 5\)
(b) increasing: \((5,\infty)\); decreasing: \((-\infty,5)\)
(c) relative maximum: DNE; relative minimum: \((5,-25)\)