identify the inflection points and local maxima and minima of the graphed function. identify the open…

identify the inflection points and local maxima and minima of the graphed function. identify the open intervals on which the function is differentiable and is concave up and concave down. find each local minimum. select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. a. there is one local minimum value of at x = (simplify your answers.) b. there are two local minima. in increasing order of x - values, the values are and at x = and x = respectively. (simplify your answers.) c. there are no local minima. find the open interval(s) on which the function is differentiable and is concave up. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the interval(s) is/are

identify the inflection points and local maxima and minima of the graphed function. identify the open intervals on which the function is differentiable and is concave up and concave down. find each local minimum. select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. a. there is one local minimum value of at x = (simplify your answers.) b. there are two local minima. in increasing order of x - values, the values are and at x = and x = respectively. (simplify your answers.) c. there are no local minima. find the open interval(s) on which the function is differentiable and is concave up. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the interval(s) is/are

Answer

Explanation:

Step1: Recall derivative - related concepts

To find local minima, we use the first - derivative test. To find concavity, we use the second - derivative test. Let (y = f(x)). First, find the first derivative (y'=f'(x)) and the second derivative (y'' = f''(x)).

Step2: Find the first derivative

If (y=\frac{1}{3}x^{3}-x^{2}-3x), then (y'=x^{2}-2x - 3=(x - 3)(x+1)) using the power rule ((x^n)'=nx^{n - 1}).

Step3: Find critical points

Set (y' = 0), so ((x - 3)(x + 1)=0). The critical points are (x=-1) and (x = 3).

Step4: Use the first - derivative test

We consider the intervals ((-\infty,-1)), ((-1,3)) and ((3,\infty)). For (x\in(-\infty,-1)), let (x=-2), then (y'=(-2 - 3)(-2 + 1)=5>0), so the function is increasing on ((-\infty,-1)). For (x\in(-1,3)), let (x = 0), then (y'=(0 - 3)(0 + 1)=-3<0), so the function is decreasing on ((-1,3)). For (x\in(3,\infty)), let (x = 4), then (y'=(4 - 3)(4 + 1)=5>0), so the function is increasing on ((3,\infty)). Since the function changes from increasing to decreasing at (x=-1), (f(-1)=\frac{1}{3}(-1)^{3}-(-1)^{2}-3(-1)=\frac{-1 - 3+9}{3}=\frac{5}{3}) is a local maximum. Since the function changes from decreasing to increasing at (x = 3), (f(3)=\frac{1}{3}(3)^{3}-(3)^{2}-3(3)=9 - 9-9=-9) is a local minimum.

Step5: Find the second derivative

(y'' = 2x-2).

Step6: Find inflection points

Set (y''=0), then (2x - 2=0), which gives (x = 1).

Step7: Determine concavity

For (x\in(-\infty,1)), let (x = 0), then (y''=2(0)-2=-2<0), so the function is concave down on ((-\infty,1)). For (x\in(1,\infty)), let (x = 2), then (y''=2(2)-2 = 2>0), so the function is concave up on ((1,\infty)).

Answer:

  1. For local minimum:
    • A. There is one local minimum value of (-9) at (x = 3).
  2. For the open - interval where the function is differentiable and concave up:
    • A. The interval(s) is ((1,\infty)).