Hi all I have homework show function $f(x)= -x^{2}$ is monotonic and show it has maximum and or minimum and don't use second derivative for this.
Please say I do wrong or not because teacher control and give me bad grade if I don't know.. pls.
$$f'(x)=-2x$$
$$f'(x)=0$$
$$0=-2x |:(-2)$$
$$x=0$$
Now check if it's minimum or maximum guys here yesterday tell me how do it and I do like they said:
For $x=-1$ we have $f(-1)=1$
For $x=0$ we have $f(0)=0$
For $x=1$ we have $f(1)=-1$
This means from $1$ it goes down to $0$ and then goes even more down to $-1$
So this is not maximum and not minimum...?
Now check monotonic:
$f'(x) > 0$ monotonic increasing in $[-1,-\infty)$
$f'(x) < 0$ monotonic decreasing in $(0,\infty)$
Is all ok or wrong please say me..
Use of derivative:
Function is increasing on interval where $f'(x)\geq 0$ and decreasing on interval where $f'(x)\leq 0$.
You can see that derivative of your function is positive on interval $x<0$ and negative on interval $x>0$. This means that function is increasing on $(-\infty, 0)$ and decreasing on $(0,\infty)$ and it is monotonic on these two intervals. However, function is $\textbf{not}$ monotonic on its whole domain.
Next, function is increasing on interval left of $x=0$ and then decreases on interval right of $x=0$. What can we conclude about that point? (without any further calculations, what does your intuition tell you)