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?\n(-2, 0)\nwhich interval contains a local minimum for this function?\n(-3, -2)\n(-1, 0)\n(0, 1)\n(0, 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?\n(-2, 0)\nwhich interval contains a local minimum for this function?\n(-3, -2)\n(-1, 0)\n(0, 1)\n(0, 2)

Answer

Explanation:

Step1: Understand local minimum concept

A local minimum is a point where the function value is less than the values at neighboring points.

Step2: Analyze function values in intervals

  • In the interval $(-3, -2)$: $f(-3)= - 20$, $f(-2)=-4$, function is increasing.
  • In the interval $(-1,0)$: $f(-1) = 0$, $f(0)=-2$, function is decreasing.
  • In the interval $(0,1)$: $f(0)=-2$, $f(1)=-4$, function is decreasing.
  • In the interval $(0,2)$: $f(0)=-2$, $f(1)=-4$, $f(2)=0$, function first decreases then increases. The function value at $x = 0$ is $-2$ and at $x=1$ is $-4$ and at $x = 2$ is $0$. The function has a local minimum in the interval $(0,1)$ as the function value at some point in this interval is less than the values at neighboring points in a small - neighborhood.

Answer:

$(0,1)$