__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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