Increasing function - A function \(f\) defined on an interval \(I\) is said to be increasing in I if \(f\left(a_{1}\right) \leq f\left(a_{2}\right)\) where \(a_{1}, a_{2} \in I\) and \(a_{1}<a_{2}\).
Strictly increasing function - A function \(f\) defined on an interval \(I\) is said to be strictly increasing on \(I\) if \(f\left(a_{1}\right)<f_{1}\left(a_{2}\right)\) where \(a_{1}, a_{2} \in I\) and also \(a_{1}<a_{2}\).
Decreasing function - A function \(f\) defined on an interval \(I\) is said to be decreasing in \(I\) if \(f\left(a_{1}\right) \geq f\left(a_{2}\right)\) where \(a_{1}, a_{2} \in I\) and \(a_{1}<a_{2}\).
Strictly decreasing function - A function \(f\) defined on an interval \(I\) is said to be strictly decreasing on \(I\) if \(f\left(a_{1}\right)>f_{1}\left(a_{2}\right)\) where \(a_{1}, a_{2} \in I\) and also \(a_{1}<a_{2}\).
We can also make use of derivative to decide whether your function over given interval \(I\) is Increasing, decreasing ,Monotone Increasing or Monotone decreasing
We can make use of above table to check what kind of nature of a given function is over a given interval \(I\).Lets try some examples.
Find the nature of function \(\;f(x)=x^2+1\;\) over \(\mathbb{R}\)
SOLUTION - First we will take its derivative ,
$$\frac{d}{dx}(x^2+1) = 2x$$
Now we will study the nature of \(\; 2x\;\) over whole \(\;\mathbb{R}\), as we can see that its zero is at \(\;x=0\), and
$$\forall x \in (0, \infty) \;\; f'(x)=2x >0$$
$$and $$
$$\forall x \in (-\infty , 0) \;\; f'(x)=2x<0$$
so form the above table we can conclude that for interval \(\;(0 ,\infty)\;\) function \(\;x^2\;\) is Strictly increasing and for interval\((-\infty,0)\) function \(\;x^2\;\) is Strictly decreasing.
Let's consider another example $$\text{Let} \; f(x)=2x^3-15x^2+36x-7\;\; \forall x \in \mathbb{R} $$
SOLUTION - First take the derivative of \(\;f(x)\)
$$f'(x) = 6x^2-30x+36$$
$$f'(x)=6(x^2-5+6)$$
we can also write this as
$$6(x-2)(x-3)$$
now we have two zeros of \(\;f'(x)\;\) , i.e at \(\;x=2\;\) and at \(\;x=3\;\) now we have to check the behavior of the derivative .
If we look carefully we will observe that for all \(\;x < 2\;\) \(f'(x)>0\)
That means function \(\;f(x)\;\) is strictly increasing for all \(\;x < 2\;\)
Again for all \(\;x >3\;\) \(f'(x)>0\)
That means function \(\;f(x)\;\) is strictly increasing for all \(\;x > 3\;\)
but for \(\; 2<x<3\;\) , \(\;f'(x) <0\)
That means function \(\;f(x)\;\) is strictly decreasing for all \(\;2<x<3\;\)
Given function \(\; f(x) = 2x^3-15x^2+36x-7\;\) is strictly increasing in \(\;(-\infty,2] \cup [3, \infty)\;\)and strictly decreasing in \(\;[2,3]\).
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