Determine the intervals on which the following function is concave up or concave

Question

Determine the intervals on which the following function is concave up or concave down. Identify any inflection points. f(x)= -x^4 -4x^3 -6x^2 -x - 7 Determine the intervals on which the following function is concave up or concave down. Select the correct choice below and fill in the answer box(es) to complete your choice. (Simplify your answer. Type your answer in interval notation. Use a comma to separate answers as needed.) A. The function is concave up on and concave down on. B. The function is concave down on. C. The function is concave up on. Locate any inflection points off. Select the correct choice below and: if necessary, fill in the answer box to complete your choice. A. An inflection point occurs at x =. (Use a comma to separate answers as needed.) B. There are no inflection points for f.

Explanation / Answer

To determine the intervals for concave up and concave down of the function we need to find the inflection points.

f(x) = -x^4-4x^3-6x^2-x-7

So to find the inflection points, first we have to find double derivative of the function.

f'(x) = -4x^3-12x^2-12x-1               [using power rule of the derivative]

f''(x) = -12x^2-24x-12

Now make f''(x) = 0 and solve for x.

so it becones 12x^2+24x + 12 = 0

Solving this I get x = -1

So there is only one inflection point.

Let me make the intervals.

Intervals (-infinity , -1 ), (-1, infinity)

If f''(x) > 0 for all x in interval I , f is concave upward in interval I.

If f''(x) < 0 for all x in interval I , f is concave down in the interval I.

Let us check the interval (-infinity , -1)

Suppose x = -2 , so the f''(x) = -12x^2-24x-12 = -12(-2)^2-24(-2)-12 = -12

So the function is concave downward in the interval (-infinity,-1)

Next, (-1, infinity)

x = 2, so the f''(x) = -12x^2-24x-12 = -12(2)^2-24(2)-12 = -108

So the function is concave downward in the interval (-1, infinity)

So the function is concave downward over all real line.

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