f(x) = 3 sin x + 3 cos x
0 ≤ x ≤ 2π
How do I use this to find local max & min, critical points, and inflection points?
I'm not sure of what you have to do, but if you have to graph f(x), you can convert it into a simple form:
$\displaystyle 3\sin x + 3 \cos x = R \sin(x + \alpha)$
$\displaystyle R = \sqrt{3^2 + 3^2} = \sqrt{9+9} = \sqrt{18}$
$\displaystyle \alpha = \tan^{-1}\left(\dfrac33\right) = \dfrac{\pi}{4}$
$\displaystyle 3\sin x + 3 \cos x = \sqrt{18} \sin\left(x + \dfrac{\pi}{4}\right)$
The first and second derivative should also be easier to graph now, provided you know how to graph f(x).
"Critical points" are, by definition, points where the first derivative is 0. "Local max and min" are points where the first derivative changes sign (and so the second derivative is not 0), and inflection points are where the second derivative is 0.
f'(x)= 3cos(x)- 3 sin(x)
That will equal 0 when cos(x)- sin(x)= 0 or cos(x)= sin(x). That happens when $\displaystyle x= \pi/4$ or $\displaystyle x= 5\pi/4$.
f''(x)= -3 sin(x)- 3cos(x)
That will equal 0 when sin(x)+ cos(x)= 0 or cos(x)=-sin(x). That happens when $\displaystyle x= 3\pi/4$ or $\displaystyle x= 7\pi/4$.
( $\displaystyle \sqrt{3^2+ 3^2}= \sqrt{2(3^2)}= 2\sqrt{3}$, not $\displaystyle \sqrt{19}$.)