An increasing function is a mathematical function that follows the pattern of increasing values as the input variable increases. This means that as the value of x increases, the value of f(x) also increases.
To prove that a function is increasing, you must show that the derivative of the function, f'(x), is always greater than or equal to 0 for all values of x. This means that the slope of the function is always positive and the function is always moving upwards.
No, a function cannot be both increasing and decreasing at the same time. It is either increasing for all values of x or decreasing for all values of x.
If a function is increasing, its graph will have a positive slope and the curve will be moving upwards. This can also be seen by examining the values of the function at different points.
Proving increasing function is important because it allows us to understand the behavior and properties of a function. It is also a fundamental concept in calculus and is used to solve various real-world problems related to rates of change and optimization.