The inflection point of a function is where the function changes concavity, i.e., where the second derivative of the function changes sign. Here is the step-by-step process to find the inflection point(s) of the function f(x) = -x^5 + 5x + 12x - 2: 1. First, find the first derivative of the function f(x). The derivative of -x^5 is -5x^4, the derivative of 5x is 5, and the derivative of 12x is 12. The derivative of -2 is 0. So, f'(x) = -5x^4 + 17. 2. Next, find the second derivative of the function f(x). The derivative of -5x^4 is -20x^3, and the derivative of 17 is 0. So, f''(x) = -20x^3. 3. Set the second derivative equal to zero and solve for x to find potential inflection points. -20x^3 = 0 implies x = 0. 4. To determine if x = 0 is an inflection point, we need to check the sign of the second derivative on either side of x = 0. If the sign changes, then x = 0 is an inflection point. 5. For x 0, f''(x) is negative. So, the sign of the second derivative does change at x = 0, and therefore, x = 0 is an inflection point. 6. To find the y-coordinate of the inflection point, substitute x = 0 into the original function f(x). f(0) = -0^5 + 5*0 + 12*0 - 2 = -2. So, the inflection point of the function f(x) = -x^5 + 5x + 12x - 2 is (0, -2). The correct answer is A. (0,-2) and (3,196).

Question

The inflection point of a function is where the function changes concavity, i.e., where the second derivative of the function changes sign.

Here is the step-by-step process to find the inflection point(s) of the function f(x) = -x^5 + 5x + 12x - 2:

1. First, find the first derivative of the function f(x). The derivative of -x^5 is -5x^4, the derivative of 5x is 5, and the derivative of 12x is 12. The derivative of -2 is 0. So, f'(x) = -5x^4 + 17.

2. Next, find the second derivative of the function f(x). The derivative of -5x^4 is -20x^3, and the derivative of 17 is 0. So, f''(x) = -20x^3.

3. Set the second derivative equal to zero and solve for x to find potential inflection points. -20x^3 = 0 implies x = 0.

4. To determine if x = 0 is an inflection point, we need to check the sign of the second derivative on either side of x = 0. If the sign changes, then x = 0 is an inflection point.

5. For x < 0, f''(x) is positive, and for x > 0, f''(x) is negative. So, the sign of the second derivative does change at x = 0, and therefore, x = 0 is an inflection point.

6. To find the y-coordinate of the inflection point, substitute x = 0 into the original function f(x). f(0) = -0^5 + 5*0 + 12*0 - 2 = -2.

So, the inflection point of the function f(x) = -x^5 + 5x + 12x - 2 is (0, -2). The correct answer is A. (0,-2) and (3,196).

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