Final answer:
To find the minimum of a function, we need to set the derivative equal to zero and solve for the x-values that satisfy the equation. Once we find these values, we can substitute them back into the original function to find the corresponding minimum values.
Step-by-step explanation:
To find the minimum of the function F(x) = 10x⁶- 48x⁵ + 15x⁴ + 200x³ - 120x² - 480x +100, we need to find the value of x that corresponds to the lowest point on the graph of the function. The minimum occurs at the x-value where the derivative of the function is equal to zero. We can find this by taking the derivative of the function and setting it equal to zero.
First, find the derivative of F(x) by applying the power rule to each term: F'(x) = 60x⁵ - 240x⁴ + 60x³ + 600x² - 240x - 480.
Set the derivative equal to zero and solve for x: 60x⁵ - 240x⁴ + 60x³ + 600x² - 240x - 480 = 0. This equation can be solved using various techniques, such as factoring, graphing, or using numerical methods like Newton's method. Once you have found the values of x that satisfy the equation, substitute them back into the original function to find the corresponding minimum values of F(x).