Question

Let $f$ be a decreasing function in $(a,b]$, then which of the following must be true?

• A. $f$ is continuous at $b$
• B. $f^{\prime }\left(b\right)<0$
• C. $\underset{x\to b}{lim}f\left(x\right)\le f\left(b\right)$
• D. $\underset{x\to b}{lim}f\left(x\right)\ge f\left(b\right)$

Answer 1

Answer: (D)

$\underset{x\to b}{lim}f\left(x\right)\ge f\left(b\right)$

$f$ is a decreasing function in $(a,b]$
So $f^{\prime }\left(x\right)\le 0$ not strictly decreasing.
Hence $f\left(b\right)\le f\left(x\right)$ in $(a,b]$
Regarding continuity, only LHS known at b, no information about RHS.