critical point, increasing/decreasing function and local extremum If $f ' (x) =(x-1)^2 (x+2)$ is given and we're asked to find the 3 things in the title:
for critical points we put $f ' (x)=0$ and the values of x we get are critical points 
for increasing/decreasing functions we will divide intervals according to critical point and take a test point to see sign of x and thus conclude if it's increasing or decreasing 
for local extremums we can use second derivative test? 
Are these three approaches correct or am i doing anything wrong?
 A: Some corrections, for technical correctness:

for critical points we put $f ' (x)=0$


*

*Critical points are where $f'(x)$

*

*either doesn't exist $(y=|x|$ at $0)$

*or equals $0.$

for local extremums we can use second derivative test



*For local extrema that are stationary turning points, the sign test is one alternative (among others) to the second-derivative test.


*But a local extremum might also be

*

*a non-stationary turning point $(y=|x|$ at $0),$ or

*a stationary non-turning point $(y=3$ at $0),$ or

*neither stationary nor turning $(y=\sqrt x$ at $0)$ !



A: If you are going to divide the interval according to whether $f$ is increasing/decreasing anyway, then you don't need the second derivative test. Just look at the monotonicity of $f$ near its critical points to decide local extremums.
In your example, $x=-2$ is a critical point of $f$. Note that $f'(x) < 0$ for $x < -2$ and $f'(x) > 0$ for $-2 < x < 1$. This means $f$ is strictly decreasing on the left of $x=-2$ and strictly increasing on the right of $x=-2$. Thus, $x=-2$ is a strict local minimum of $f$.
You can do a similar analysis for another critical point $x=1$.
A: Yes, you are right!
The first derivative test uses the derivatives of a function to locate the critical points of a function. To calculate the critical points we need to solve the equation $f^{'}(x)=0$. The computed values of the parameter $x$ will be critical.
The second-derivative test uses the value of the second derivative at those points to determine whether such points are a local maximum or a local minimum. If the function f is twice-differentiable at a critical point x (i.e. a point where $f^{'}(x)=0$), then:

*

*If $f^{''}(x)<0$, then f has a local maximum at x.


*If $f^{''}(x)>0$, then f has a local minimum at x.


*If $f^{''}(x)=0$, the test is inconclusive.
Good luck!
