Determine the x-value for each inflection point on the graph of the following function.

f(x)=3x^5-5x^4-80x^3+360x^2+1000x+850

inflection at y'' = 0

y' = 15x^4 - 20x^3 - 240x^2 + 720x + 1000

y'' = 60x^3 - 60x^2 - 480x + 720

so solve y'' = 0

x^3 - x^2 - 8x + 12 = 0

(x-2)(x-2)(x+3) = 0

There you have the x values. I trust you can evaluate y(x) to get the rest.

so would the inflection point be x=-3 because that's where it goes from decreasing to increasing

yes

To determine the x-values for the inflection points of the function f(x) = 3x^5 - 5x^4 - 80x^3 + 360x^2 + 1000x + 850, you need to find the second derivative of the function and solve for the points where it equals zero.

Step 1: Find the first derivative of the function f(x):
f'(x) = 15x^4 - 20x^3 - 240x^2 + 720x + 1000

Step 2: Find the second derivative of the function f'(x):
f''(x) = 60x^3 - 60x^2 - 480x + 720

Step 3: Set f''(x) equal to zero and solve for x:
60x^3 - 60x^2 - 480x + 720 = 0

This equation doesn't have a nice factoring solution, so you would need to use numerical methods, such as Newton's method or the bisection method, to find the approximate solutions. Alternatively, you can use software or graphing calculators that have built-in functions to solve equations.

Once you have found the solutions to the equation f''(x) = 0, those x-values will correspond to the inflection points on the graph of the function f(x).