For all real values of x, the minimum value of f(x) = (x 2 + 2x – 15) x is;

To find the minimum value of the function f(x) = (x^2 + 2x - 15)x, we first need to find its derivative. The derivative of a function gives us the slope of the function at any point, and the points where the derivative is zero are the points where the function reaches a maximum or minimum.

The derivative of f(x) = (x^2 + 2x - 15)x is found using the product rule, which states that the derivative of two functions multiplied together is the first function times the derivative of the second, plus the second function times the derivative of the first.

So, the derivative of f(x) = (x^2 + 2x - 15)x is:

f'(x) = (x^2 + 2x - 15) * 1 + x * (2x + 2)

Simplifying this gives:

f'(x) = x^2 + 2x - 15 + 2x^2 + 2x

Combining like terms gives:

f'(x) = 3x^2 + 4x - 15

Setting this equal to zero gives the x-values where the function reaches a maximum or minimum:

3x^2 + 4x - 15 = 0

Solving this quadratic equation will give the x-values where the function reaches a maximum or minimum. From there, you can substitute these x-values back into the original function to find the corresponding y-values, and the smallest of these y-values will be the minimum value of the function.