Find the absolute maximum and minimum, if either exists, for \( f(x)=x^{2}-10x+5 \) Find the first derivative of \( f \). \[ f'(x)=2x-10 \] Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The absolute minimum is \( \square \) at \( x=\square \). B. There is no absolute minimum.

Set the first derivative equal to zero to find the critical points.

\(2x - 10 = 0\)

Solve the equation \(2x - 10 = 0\) for \(x\).

\(x = 5\)

Plug the critical point \(x = 5\) into the original function \(f(x) = x^2 - 10x + 5\) to find the corresponding y-value.

\(f(5) = 5^2 - 10(5) + 5 = 25 - 50 + 5 = -20\)

Determine if the critical point \(x = 5\) represents an absolute minimum by considering the second derivative or the nature of the function.

Since \(f(x) = x^2 - 10x + 5\) is a parabola opening upwards (because the coefficient of \(x^2\) is positive), the critical point \(x = 5\) represents an absolute minimum.

Conclude the absolute minimum value and the \(x\) value at which it occurs.

The absolute minimum is \(-20\) at \(x = 5\).