find the absolute extrema of the function (if any exist) on each interval. (if an answer does not exist…

find the absolute extrema of the function (if any exist) on each interval. (if an answer does not exist, enter dne.\n$f(x)=x^{2}-8x$\n(a) $-1,8$\nminimum $(x,y)=$\nmaximum $(x,y)=$\n(b) $(4,9$\nminimum $(x,y)=$\nmaximum $(x,y)=$\n(c) $(0,8)$\nminimum $(x,y)=$\nmaximum $(x,y)=$\n(d) $4,10)$\nminimum $(x,y)=$\nmaximum $(x,y)=$

find the absolute extrema of the function (if any exist) on each interval. (if an answer does not exist, enter dne.\n$f(x)=x^{2}-8x$\n(a) $-1,8$\nminimum $(x,y)=$\nmaximum $(x,y)=$\n(b) $(4,9$\nminimum $(x,y)=$\nmaximum $(x,y)=$\n(c) $(0,8)$\nminimum $(x,y)=$\nmaximum $(x,y)=$\n(d) $4,10)$\nminimum $(x,y)=$\nmaximum $(x,y)=$

Answer

Explanation:

Step1: Find the derivative of the function

The function is (f(x)=x^{2}-8x). Using the power rule ((x^n)^\prime = nx^{n - 1}), the derivative (f^\prime(x)=2x - 8).

Step2: Find the critical points

Set (f^\prime(x)=0), so (2x-8 = 0). Solving for (x) gives (x = 4).

Part (a) Interval ([-1,8])

Step1: Evaluate the function at critical and endpoints

  • (f(-1)=(-1)^{2}-8\times(-1)=1 + 8=9)
  • (f(4)=4^{2}-8\times4=16-32=-16)
  • (f(8)=8^{2}-8\times8=64 - 64=0)

Part (b) Interval ((4,9])

Step1: Analyze the behavior of the function

Since (f^\prime(x)=2x - 8), for (x>4), (f^\prime(x)>0) (function is increasing).

  • (f(4)) is not in the open - ended part of the interval ((4,9]). Evaluate (f(9)=9^{2}-8\times9=81-72 = 9)

Part (c) Interval ((0,8))

Step1: Analyze the behavior of the function

Since (f^\prime(x)=2x - 8), (f(x)) is decreasing on ((0,4)) (because (f^\prime(x)<0) when (x\in(0,4))) and increasing on ((4,8)) (because (f^\prime(x)>0) when (x\in(4,8))). But the endpoints (x = 0) and (x = 8) are not included.

Part (d) Interval ([4,10))

Step1: Analyze the behavior of the function

Since (f^\prime(x)=2x - 8), for (x\geq4), (f^\prime(x)\geq0) (function is non - decreasing). (f(4)=4^{2}-8\times4=-16) and (f(10)=10^{2}-8\times10=100 - 80 = 20) (but (x = 10) is not included)

Answer:

(a) minimum ((x,y)=(4,-16)) maximum ((x,y)=(-1,9))

(b) minimum DNE maximum ((x,y)=(9,9))

(c) minimum DNE maximum DNE

(d) minimum ((x,y)=(4,-16)) maximum DNE