This is first part of multipart q3

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- $f$ is said to have a maximum value in $I$ , if there exist a point c in I such that $f(c) \geq f (x)$ for all $x \in I$.The number $f( c)$ is called the maximum value of f in I and the point c is called a point of maximum value of f in I
- $f$ is said to have a minimum value in $I$ , if there exist a point $c$ in I such that $f(c) \leq f (x)$ for all $x \in I$.The number $f(c)$ is called the minimum value of f in I and the point $c$ in this case is called a point of minimum value of $f$ in I
- $f$ is said to have a extreme value in $I$ , if there exist a point $c$ in I such that f(c) is either a maximum value or minimum value of $f$ in $I$. The number $f (c)$ in this case is called the extreme value of $f$ in $I$ and the point $c$ is called the extreme point.

Step 1:

$f(x)=x^2$

Let $f(x)=x^2$

$f'(x)=2x$[Differentiating with respect to x]

Now $f'(x)=0$

$\Rightarrow 2x=0$

i.e $x=0$

Step 2:

At $x=0$

When $x$ is slightly < 0$\Rightarrow f'(x)$ is -ve.

When $x$ is slightly > 0$\Rightarrow f'(x)$ is +ve.

Therefore $f'(x)$ changes sign from -ve to +ve as $x$ increases through 0

$\Rightarrow f(x)$ has a local minimum at $x=0$.

Local minimum values =$f(0)=0$

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