just reviewing something from calc 1
(made up ex.)

Question

just reviewing something from calc 1
(made up ex.)

idku · Answer

Should I make a function up?
$\tt f(x)=x^3+3x^2+2x+1$

now will do a lot of things to it...

idku · Answer

At first, I will just simply determine when the function is increasing and when decreasing, using the first derivative.

idku · Answer

$\tt f'(x)=3x^2+6x+2$
This is the derivative, or the slope, of the function $\tt f(x)$.

We know from elementary maths before calculuses, that:
$\bullet$ When the slope is positive the function is increasing
$\bullet$ When the slope is negative the function is decreasing

idku · Answer

So, we will set  $\tt f'(x)>0$  to find where the function is increasing, and set $\tt f'(x)<0$ to find where the function is decreasing.

idku · Answer

now, won't actually solve that... i need the concepts.

idku · Answer

Ok, now, the critical numbers.

idku · Answer

the critical numbers are the values of $\tt f(x)$, are when:

1.   $\tt f'(x)=0$
(will have to set the derivative =0 and solve for x)

2.   $\tt f'(x)$ is undefined (provided that $\tt f(x)$ has a value at that point)
Polynomial is always continuous, and even without knowing this fact, I can tell that no restrictions in $\tt f(x)$ or $\tt f'(x)$ exist.

idku · Answer

Then
$\tt f'(b)=greatest~output~of~B$
$\tt f'(a)=smallest~output~of~A$

when i check critical numbers, that would mean that I have
absolute maximum of B at x=b
absolute minimum of A at x=a