determine the location and value of the absolute extreme values of f on the given interval, if they exist\n…

determine the location and value of the absolute extreme values of f on the given interval, if they exist\n f(x)=\frac{12 x^{3}}{3}+10 x^{2}-8 x \text { on }-3,1 \nwhat is/are the absolute maximum/maxima of f on the given interval? select the correct choice below and, if necessary, fill in the answer boxes to complete your choice\na. the absolute maximum/maxima is/are 24 at ( x=-2 )\n(use a comma to separate answers as needed. type exact answers, using radicals as needed)\nb. there is no absolute maximum of f on the given interval\nwhat is/are the absolute minimum/minima of f on the given interval? select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.\na. the absolute minimum/minima is/are at ( x= )\n(use a comma to separate answers as needed. type exact answers, using radicals as needed)\nb. there is no absolute minimum of f on the given interval

determine the location and value of the absolute extreme values of f on the given interval, if they exist\n f(x)=\frac{12 x^{3}}{3}+10 x^{2}-8 x \text { on }-3,1 \nwhat is/are the absolute maximum/maxima of f on the given interval? select the correct choice below and, if necessary, fill in the answer boxes to complete your choice\na. the absolute maximum/maxima is/are 24 at ( x=-2 )\n(use a comma to separate answers as needed. type exact answers, using radicals as needed)\nb. there is no absolute maximum of f on the given interval\nwhat is/are the absolute minimum/minima of f on the given interval? select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.\na. the absolute minimum/minima is/are at ( x= )\n(use a comma to separate answers as needed. type exact answers, using radicals as needed)\nb. there is no absolute minimum of f on the given interval

Answer

Explanation:

Step1: Simplify the function

Simplify ( f(x)=\frac{12x^{3}}{3}+10x^{2}-8x ) to ( f(x) = 4x^{3}+10x^{2}-8x ).

Step2: Find the derivative

Differentiate ( f(x) ) using the power rule ( (x^{n})^\prime=nx^{n - 1} ). ( f^\prime(x)=12x^{2}+20x - 8 ). Factor ( f^\prime(x) ): ( f^\prime(x)=4(3x^{2}+5x - 2)=4(3x - 1)(x + 2) ).

Step3: Find critical points

Set ( f^\prime(x)=0 ), then ( 4(3x - 1)(x + 2)=0 ). Solving ( 3x-1 = 0 ) gives ( x=\frac{1}{3} ), and solving ( x + 2=0 ) gives ( x=-2 ). Both ( x=-2) and (x = \frac{1}{3}) are in the interval ([-3,1]).

Step4: Evaluate the function at critical points and endpoints

  • Evaluate ( f(x) ) at ( x=-3): ( f(-3)=4\times(-3)^{3}+10\times(-3)^{2}-8\times(-3)=4\times(-27)+10\times9 + 24=-108 + 90+24 = 6 ).
  • Evaluate ( f(x) ) at ( x=-2): ( f(-2)=4\times(-2)^{3}+10\times(-2)^{2}-8\times(-2)=4\times(-8)+10\times4+16=-32 + 40+16=24 ).
  • Evaluate ( f(x) ) at ( x=\frac{1}{3}): ( f(\frac{1}{3})=4\times(\frac{1}{3})^{3}+10\times(\frac{1}{3})^{2}-8\times\frac{1}{3}=4\times\frac{1}{27}+10\times\frac{1}{9}-\frac{8}{3}=\frac{4 + 30-72}{27}=-\frac{38}{27}\approx - 1.41 ).
  • Evaluate ( f(x) ) at ( x = 1): ( f(1)=4\times1^{3}+10\times1^{2}-8\times1=4 + 10-8 = 6 ).

Answer:

The absolute maximum is (24) at (x=-2). The absolute minimum is (-\frac{38}{27}) at (x=\frac{1}{3}). So for the first question, the answer is A. The absolute maximum/maxima is/are (24) at (x=-2). For the second question, the answer is A. The absolute minimum/minima is/are (-\frac{38}{27}) at (x = \frac{1}{3}).