Q.
If the function $f$ be given by $f ( x )= x ^{3}-3 x +3$, then
I. $x=\pm 2$ are the only critical points for local maxima or local minima.
II. $x =1$ is a point of local minima.
III. local minimum value is $2$ .
IV. local maximum value is $5$ .
Application of Derivatives
Solution:
We have $f(x)=x^{3}-3 x+3$
or $f'( x )=3 x ^{2}-3=3( x -1)( x +1)$
or $f'( x )=0$ at $x =1$ and $x =-1$
Thus, $x =\pm 1$ are the only critical points which could possibly be the points of local maxima and/or local minima of $f$. Let us first examine the point $x =1$.
Note that for values close to $1$ and to the right of $1 . f$ $'(x) > 0$ and for values close to $1$ and to the left of $1 . f$ $'(x) < 0 .$
Therefore, by first derivative test, $x=1$ is a point of local minima and local minimum value is $f (1)=1 .$
In the case of $x =-1$, note that $f '( x )>0$, for values close to and to the left of $-1$ and $f '( x ) < 0$, for values close to and to the right of $-1$.
Therefore, by first derivative test, $x =-1$ is a point of local maxima and local maximum value is $f (-1)=5$
Values of x
Sign of $f'(x) = 3(x - 1)(x + 1)$
Close to 1
to the right (say 1 .1 etc.)
> 0
to the left (say 0.9 etc.)
< 0
Close to -1
to the right (say -0.9 etc.)
< 0
to the left (say -1.1 etc.)
> 0
| Values of x | Sign of $f'(x) = 3(x - 1)(x + 1)$ | |
|---|---|---|
| Close to 1 | to the right (say 1 .1 etc.) | > 0 |
| to the left (say 0.9 etc.) | < 0 | |
| Close to -1 | to the right (say -0.9 etc.) | < 0 |
| to the left (say -1.1 etc.) | > 0 | |