ISI2016-MMA-24

Question

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a strictly increasing function. Then which one the following is always true?

• A. The limits $\lim_{x \rightarrow a+} f(x) $ and $\lim_{x \rightarrow a-} f(x)$ exist for all real numbers $a$
• B. If $f$ is differentiable at $a$ then $f'(a)>0$
• C. There cannot be any real number $B$ such that $f(x)<B$ for all real $x$
• D. There cannot be any real number $L$ such that $f(x)>L$ for all real $x$

Answer 1

The statement claims that a strictly increasing function, f, has the following properties:

• The limits, lima→+∞​f(x) and lima→−∞​f(x), exist for all real numbers, a. (Choice A)
• If f is differentiable at a, then f′(a)>0. (Choice B)
• There cannot be any real number, B, such that f(x)<B for all real numbers, x. (Choice C)
• There cannot be any real number, L, such that f(x)>L for all real numbers, x. (Choice D)

Of the above choices, only Choice B is always true for strictly increasing functions.

Let's analyze the other options:

• Choice A: Consider the function f(x)=x2. As x approaches positive or negative infinity, the function also approaches positive infinity, so the limits don't exist. This function is strictly increasing.
• Choice C: Consider the function f(x)=x+1. This function is strictly increasing, but for any negative real number, B, the inequality f(x)<B will always hold.
• Choice D: This is similar to Choice C. Consider the function f(x)=−x−1. This function is strictly increasing, but for any positive real number, L, the inequality f(x)>L will always hold.

Therefore, the only option that is always true for strictly increasing functions is B), if f is differentiable at a, then f'(a) > 0.