Question

What is the meaning of monotonically increasing function?

Answer 1

Hint: This type of question depends on the concept of monotonic functions, these concepts are helpful when studying exponential and logarithmic functions. A function is said to be monotonic if its derivative does not change sign. We know that the groups of monotonically increasing and monotonically decreasing functions have some special properties. Also we know that a monotonically function is one that increases as x does for all real x.

Complete step by step solution:
Now, here we have to write the meaning of monotonically increasing function.
Let us suppose a function is continuous on [a, b] and it is differentiable on (a, b). If the derivative is larger than zero for all x in (a, b), then the function is monotonically increasing on [a, b].
We can represent a monotonically increasing function as \[{{x}_{0}} < {{x}_{1}}\Rightarrow f\left( {{x}_{0}} \right) < f\left( {{x}_{1}} \right)\].
We can write the meaning of monotonically increasing function as that we have a function with positive slope in every point of domain. Starting from a point \[{{x}_{0}}\] and moving to the right, the graph of the function is also moving up at the same time.
If \[f\left( x \right)\] is a monotonically increasing function then there is only one solution as no value of \[f\left( x \right)\] is achieved twice.

Note: In this type of this question students may make mistakes in the value of \[f\left( x \right)\]. Students have to take note that over an interval on which a function is monotonically increasing, an output for the function will not occur more than once.