How to find the range of polynomial functions without knowing extrema?

[skeptipus]

Hello

I'm asked to find the range of this function:


It's easy to find the range if you are provided a graph, or if you are allowed to use a graphing calculator.

But I'd have to draw graphs by hand in final exam.

I've not been taught yet how to find minimums and maximums.

So, my question is how can I determine the range of a function without knowing the extrema?

The range of the function is

and (4.868, 499.82) is the maximum point (I hope I'm using the correct term here).

 

[pka]

I'm asked to find the range of this function:
View attachment 10677
But I'd have to draw graphs by hand in final exam.
I've not been taught yet how to find minimums and maximums.
So, my question is how can I determine the range of a function without knowing the extrema?
The range of the function is
View attachment 10678
and (4.868, 499.82) is the maximum point (I hope I'm using the correct term here).

Let's rewrite \(\displaystyle f(x)=-5(2x+1)(x-2)^2(x-6)\)
If \(\displaystyle x\in(-\infty,-0.5)\cup(6,\infty)\) then \(\displaystyle f(x)<0\). Is that correct?
Now if \(\displaystyle x\in(-0.5,6)\) then \(\displaystyle f(x)\ge 0\), do you follow that?
How big can \(\displaystyle f(x)\) become on that open interval? Let's say \(\displaystyle x=5\). Is it \(\displaystyle -5(11)(3)^2(-1)=495~?\)

Can you carry on?

 

[skeptipus]

Let's rewrite \(\displaystyle f(x)=-5(2x+1)(x-2)^2(x-6)\)
If \(\displaystyle x\in(-\infty,-0.5)\cup(6,\infty)\) then \(\displaystyle f(x)<0\). Is that correct?
Now if \(\displaystyle x\in(-0.5,6)\) then \(\displaystyle f(x)\ge 0\), do you follow that?
How big can \(\displaystyle f(x)\) become on that open interval? Let's say \(\displaystyle x=5\). Is it \(\displaystyle -5(11)(3)^2(-1)=495~?\)

Can you carry on?

Oh!! :shock: I see your reasoning! Thank you very much, sir!