Question

The function f(x)=−2x³+45x²−300x+10 has one local minimum and one local maximum. This function has a local minimum at x=______ with a value of f(x)=______, and a local maximum at x =______ with a value of f(x)=______.

Answer 1

The local minimum and local maximum of the function f(x) = -2x³ + 45x² - 300x + 10 are x = 5 and x = 15, with values of f(x) equal to -1850 and 69260, respectively.

Step-by-step explanation:

The local minimum and local maximum of the function f(x) = -2x³ + 45x² - 300x + 10 are:

Local minimum at x = 5 with a value of f(x) = -1850

Local maximum at x = 15 with a value of f(x) = 69260

To find the local minimum and local maximum, we need to find the critical points of the function. First, we find the derivative of the function: f'(x) = -6x² + 90x - 300. Set f'(x) equal to zero and solve for x to find the critical points. Then, use the second derivative test to determine whether the critical points are local minimum or local maximum.