If f(x) is defined as follows, what is the minimum value of f(x) for x ∊ (0,…

To find the minimum value of f(x) on (0, 2], analyze both pieces of the piecewise function.

For 0 < x <= 1.5, f(x) = 25/(8x), which decreases as x increases. Hence the minimum in this interval occurs at x = 1.5:
f(1.5) = 25/(8 x 1.5) = 25/12 = 2 + 1/12.

For 1.5 < x <= 2, f(x) = x + 1/x. Its derivative is 1 - 1/x², which is positive for x > 1. Thus this piece is increasing on (1.5, 2], so its least limiting value near 1.5 is 1.5 + 1/1.5 = 13/6 = 2 + 1/6, which is larger than 25/12.

Therefore, the minimum value of f(x) is 25/12 = 2 + 1/12.