Question

Which of the following can be said true for a function $f$ to be increasing at $x_0$, in an interval $I=\left(x_0-h, x_0+h\right), h>0$ such that $x_1, x_2 \in R$ ?

• A. $x_1 < x_2$ in $I \Rightarrow f\left(x_1\right) \leq f\left(x_2\right)$
• B. $x_1>x_2$ in $I \rightarrow f\left(x_1\right) \leq f\left(x_2\right)$
• C. $x_1 < x_2$ in $I \Rightarrow f\left(x_1\right) \geq f\left(x_2\right)$
• D. None of the above

Answer 1

Answer: (A)

For a function to be increasing or decreasing at a point, we follow the definition given below.
If $x_0$ be a point in the domain of definition of a real valued function $f$. Then, $f$ is said to be increasing, strictly increasing, decreasing or strictly decreasing at $x_0$ if there exists an open interval $I$ containing $x_0$ such that $f$ is increasing, strictly increasing, decreasing or strictly decreasing, respectively, in/.
Let us clarify this definition for the case of increasing function.
A function $f$ is said to be increasing at $x_0$ if there exists an interval $I=\left(x_0-h, x_0+h\right), h>0$ such that for $x_1, x_2 \in I$
$x_1 < x_2 \text { in I } \Rightarrow f\left(x_1\right) \leq f\left(x_2\right)$
Similarly, the other cases can be clarified.