If `f(x)ge 0` and `g(x)ge 0` for all real x, which of the following statements must be true ?
I. `f(x)+g(x)ge 0`
II. `f(x)-g(x)ge 0`
III. `f(x)g(x)ge 0`

Options

A. I only
B. II only
C. III only
D. I and III

Answer

D

Explanation

Don't let the functions symbolism confuse you. Just think of these as two quantities - f(x) and g(x) - that are both nonnegative. Statement I says their sum is nonnegative. That's true - add any two nonnegatives and you'll get a nonnegative sum. Statement II says the difference `f(x)-g(x)` is nonnegative. Well, that's true only if `f(x) ge g(x)`. But there's no reason that g(x) couldn't be greater than f(x). Statement III says their product is nonnegative. That's true - multiply any two nonnegatives and you'll get a nonnegative product. Statements I and III are true.