Increasing and Decreasing Functions: Overview, Questions, Preparation

If the output y increases as along with increase in the input x, the function is increasing. And it is decreasing if its output y decreases with increase in input x.

Let us understand this briefly through mathematical equations.

A function f(x) is considered to be increasing if:

f(x1)≤f(x2)

for any two numbers x1,x2∈ I with x1

Similarly, A function f(x)f(x)f(x) is said to be decreasing if:

f(x1​)≥f(x2​)

1. Constant Functions: This is referred to the function that does not change with the input (f(x) = constant). 
2. Derivatives: The derivative of a function is a term that is used to provide information about its increasing/decreasing status at a particular point of time.
3. Critical Points: Critical points are the points where f'(x) = 0 or undefined. These points are an indicator of where a function changes from increasing to decreasing or vice versa. 
4. Testing Interval: Testing Interval is to check the sign of the derivative at different points of time to know when and where the function is increasing or decreasing.

Question: Find the critical points for the function f(x)=x^2−4x.

Answer:

Critical points for the function: 2x−4=0⇒x=2

Similarly, intervals: (−∞,2)(2,∞)

Now, coming to the testing part.

For x=0, f′(0)=−4 <0 → decreasing.

For x=3, f′(3)=2 >0→ increasing.

Hence, the function increases at (2,∞) and decreases at (−∞,2).