find the exact location of all the relative and absolute extrema of the function (f(x)=3x^{2}-6x - 9) with…

find the exact location of all the relative and absolute extrema of the function (f(x)=3x^{2}-6x - 9) with domain (0,3). (f) has ? at ((x,y)=) (smallest (x - value)) (f) has ? at ((x,y)=) (largest (x - value))

find the exact location of all the relative and absolute extrema of the function (f(x)=3x^{2}-6x - 9) with domain (0,3). (f) has ? at ((x,y)=) (smallest (x - value)) (f) has ? at ((x,y)=) (largest (x - value))

Answer

Explanation:

Step1: Find the derivative of the function

The function is (f(x)=3x^{2}-6x - 9). Using the power - rule ((x^n)^\prime=nx^{n - 1}), we have (f^\prime(x)=6x-6).

Step2: Set the derivative equal to zero and solve for (x)

Set (f^\prime(x)=0), so (6x - 6=0). Add 6 to both sides: (6x=6), then divide by 6 to get (x = 1).

Step3: Evaluate the function at the critical point and endpoints

The endpoints of the domain ([0,3]) are (x = 0) and (x = 3), and the critical point is (x = 1).

  • When (x = 0), (f(0)=3\times0^{2}-6\times0 - 9=-9).
  • When (x = 1), (f(1)=3\times1^{2}-6\times1 - 9=3 - 6 - 9=-12).
  • When (x = 3), (f(3)=3\times3^{2}-6\times3 - 9=27-18 - 9=0).

Answer:

(f) has the smallest (y - value) at ((x,y)=(1,-12)) (smallest (x - value) among the extrema is (x = 1)), (f) has the largest (y - value) at ((x,y)=(3,0)) (largest (x - value) among the extrema is (x = 3))