find the absolute maximum and absolute minimum values of the function\n\n$f(x)=x^{3}+12x^{2}-27x +…

find the absolute maximum and absolute minimum values of the function\n\n$f(x)=x^{3}+12x^{2}-27x + 9$\n\nover each of the indicated intervals.\n\n(a) the interval $=-10,0$.\n\n1. absolute maximum $=504$\n\n2. absolute minimum $=9$\n\n(b) the interval $=-7,2$.\n\n1. absolute maximum $=443$\n\n2. absolute minimum $=-5$\n\n(c) the interval $=-10,2$.\n\n1. absolute maximum $=504$\n\n2. absolute minimum $=-5$

find the absolute maximum and absolute minimum values of the function\n\n$f(x)=x^{3}+12x^{2}-27x + 9$\n\nover each of the indicated intervals.\n\n(a) the interval $=-10,0$.\n\n1. absolute maximum $=504$\n\n2. absolute minimum $=9$\n\n(b) the interval $=-7,2$.\n\n1. absolute maximum $=443$\n\n2. absolute minimum $=-5$\n\n(c) the interval $=-10,2$.\n\n1. absolute maximum $=504$\n\n2. absolute minimum $=-5$

Answer

Explanation:

Step1: Find the derivative of the function

The derivative of (f(x)=x^{3}+12x^{2}-27x + 9) is (f^{\prime}(x)=3x^{2}+24x-27 = 3(x^{2}+8x - 9)=3(x + 9)(x-1))

Step2: Find the critical points

Set (f^{\prime}(x)=0), then (3(x + 9)(x - 1)=0). The critical points are (x=-9) and (x = 1)

Step3: Evaluate the function at critical points and endpoints for interval ([-10,2])

  • For (x=-10): (f(-10)=(-10)^{3}+12(-10)^{2}-27(-10)+9=-1000 + 1200+270 + 9=479)
  • For (x=-9): (f(-9)=(-9)^{3}+12(-9)^{2}-27(-9)+9=-729+972 + 243+9=504)
  • For (x=1): (f(1)=1^{3}+12\times1^{2}-27\times1 + 9=1+12-27 + 9=-5)
  • For (x = 2): (f(2)=2^{3}+12\times2^{2}-27\times2+9=8 + 48-54+9=11)

Answer:

  1. Absolute maximum = (504)
  2. Absolute minimum=(-5)