x -4 -3 -2 -1 0 1 2 3 4\nf(x) -54 -20 -4 0 -2 -4 0 16 50\nwhich interval contains a local maximum for this…

x -4 -3 -2 -1 0 1 2 3 4\nf(x) -54 -20 -4 0 -2 -4 0 16 50\nwhich interval contains a local maximum for this function?\nwhich interval contains a local minimum for this function?\n(-3, -2)\n(-2, 0)\n(0, 1)\n(1, 2)

x -4 -3 -2 -1 0 1 2 3 4\nf(x) -54 -20 -4 0 -2 -4 0 16 50\nwhich interval contains a local maximum for this function?\nwhich interval contains a local minimum for this function?\n(-3, -2)\n(-2, 0)\n(0, 1)\n(1, 2)

Answer

Explanation:

Step1: Recall local - maximum definition

A local maximum occurs when the function value is greater than the function values at nearby points.

Step2: Analyze function values in intervals

  • In the interval ((-3,-2)): (f(-3)= - 20), (f(-2)=-4), the function is increasing.
  • In the interval ((-2,0)): (f(-2)=-4), (f(0)= - 2), the function is increasing from (x=-2) to (x = 0).
  • In the interval ((0,1)): (f(0)=-2), (f(1)=-4), the function is decreasing. Since the function changes from increasing to decreasing in the interval ((0,1)), there is a local maximum in this interval.
  • In the interval ((1,2)): (f(1)=-4), (f(2)=0), the function is increasing.

Answer:

((0,1))