consider the following function. use a graphing utility to confirm your answers for parts (a) through (c)…

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.)\n\n$f(x)=x^{2}-10x$\n\n(a) find the critical numbers of f. (enter your answers as a comma-separated list.)\n\n$x=$\n\n(b) find the open intervals on which the function is increasing or decreasing. (enter your answers using interval notation.)\n\nincreasing\n\ndecreasing\n\n(c) apply the first derivative test to identify all relative extrema.\n\nrelative maximum $(x,y)=(quad)$\n\nrelative minimum $(x,y)=(quad)$

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.)\n\n$f(x)=x^{2}-10x$\n\n(a) find the critical numbers of f. (enter your answers as a comma-separated list.)\n\n$x=$\n\n(b) find the open intervals on which the function is increasing or decreasing. (enter your answers using interval notation.)\n\nincreasing\n\ndecreasing\n\n(c) apply the first derivative test to identify all relative extrema.\n\nrelative maximum $(x,y)=(quad)$\n\nrelative minimum $(x,y)=(quad)$

Answer

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): [ \begin{align*} 2x&=10\ x& = 5 \end{align*} ]

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))