To find where f(x) obtains its minimum value, we first need to find the derivative of the function. The derivative will give us the rate of change of the function, and for a minimum point, this rate of change should be equal to zero. A minimum will occur where the slope of the tangent line to the function is zero, i.e., where the derivative is zero.
The given function is f(x) = 5x² - 4x - 8. Let's find the derivative (f'(x)).
The derivative of 5x² in respect to x is 10x (Using the power rule: n*x^n-1, where n=2 in 5x² gives us 10x).
The derivative of -4x in respect to x is -4.
The derivative of -8 in respect to x is 0 as the derivative of any constant is 0.
So the derivative of the function f(x) is f'(x) = 10x - 4.
To find where this derivative is equal to zero, we can set f'(x) equal to zero and solve for x:
10x - 4 = 0
Add 4 to both sides to isolate the term with x:
10x = 4
Then divide both sides by 10 to solve for x:
x = 4 / 10 = 0.4 or 2/5.
Hence, f(x) obtains its minimum value at x = 2/5.