Question

Find the intervals in which f is increasing or decreasing for F(x)=sinx+cosx 0 12 years ago

Answer 1

\[\Large\bf f(x)\quad=\quad \sin x +\cos x\]\[\Large\bf f'(x)\quad=\quad ?\]

Answer 2

Cosx-sinx

Answer 3

Mmm k cool, looks good so far.
So umm we want to find critical points, then we can determine what's happening between each critical point (whether the function is increasing or decreasing).

To find critical points we set the first derivative equal to zero and solve for x.\[\Large\bf 0\quad=\quad \cos x - \sin x\]Ok sooooo, understand how to solve for x from this point? :o

Answer 4

No that's the part I don't understand :c

Answer 5

Let's add sinx to each side,\[\Large\bf \sin x\quad=\quad \cos x\]Then we'll divide each side by cos x,\[\Large\bf\frac{\sin x}{\cos x}\quad=\quad 1\]

Answer 6

Recall your trig identity:\[\Large\bf \frac{\sin x}{\cos x}\quad=\quad tan x\]

Answer 7

So we currently have:\[\Large\bf \tan x\quad=\quad 1\]Remember your unit circle?
What values give us 1 from tangent?

Answer 8

Pi/4?

Answer 9

Mmm good!
There's another one though.

Answer 10

Hmm 5pi/4?

Answer 11

Ummm yes, that's the one!

Answer 12

Setting up a number line for our critical points:

Answer 13

We want to look at test points on each side of our critical points to see what's happening.

So we're going to plug test points into our derivative function.
If the derivative gives us a (positive value), then the function is increasing over that interval.
If negative, then decreasing.

Answer 14

So if we want to test a point smaller than pi/4, let's try using pi/6.