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# Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: $(i)\: f (x) = x^2$

This is first part of multipart q3

Toolbox:
• $f$ 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 =$f(0)=0$