# Thread: local max. and min.

1. ## local max. and min.

Find a formula for a function of the form y = Acos(Bx) + C with a maximum of (0 , 9), a minimum of (5 , 6) and no critical points between these two points.
A=
B=
C=

need some help with this bad johnny, dont know where to begin...thanks

mathaction

2. Originally Posted by mathaction
Find a formula for a function of the form y = Acos(Bx) + C with a maximum of (0 , 9), a minimum of (5 , 6) and no critical points between these two points.
...
Hello,

1. calculate the 1rst derivative (use chain rule!):

$f'(x) = -AB\cdot \sin(Bx)$

You know that f'(0) = 0 and f'(5) = 0

Plug in these values into the equation of the derivative:

$f'(0)=0=-AB\cdot \sin(0)$ that means you can't determine the values of the parameters.

$f'(5) = 0 = -AB\cdot \sin(5B)~\iff~\sin(5B) = 0~\iff~5B=k\cdot \pi, k\in \mathbb{Z}~\iff~$ $B=k\cdot \frac \pi5$

Since you know that at x = 5 is the next extremum you know that in this case k = 1 so $B=\frac \pi5$

2. Plug in the coordinates of the points you know into the equation of the function:

$f(0)=9=A\cdot \cos\left(\frac \pi5 \cdot 0 \right)+C~\implies~\boxed{9=A+C}$ because $\cos(0) = 1$

$f(5) = 6=A\cdot \cos\left(\frac \pi5 \cdot 5 \right)+C~\implies~\boxed{6=-A+C}$ because $\cos(\pi) = -1$

3. Now you have a system of 2 simultaneous equations. Solve for A and C.

I've got A = 1.5 and C = 7.5

$f(x) = 1.5 \cdot cos\left(\frac \pi5 \cdot x\right)+7.5$

I've attached a diagram of the graph.

3. y = A*cos(Bx) +C

Max at (0,9) and min at (5,6)

That means the ampltude A is (1/2)(9-6) = 1.5

It means also that the perod is 2(5-0) = 10
So, 10 = 2pi/B
B = 2pi/10 = pi/5

It means also that the netral axis s at 6 +1.5 = 7.5
Meaning, the basic cosine curve is shifted vertically 7.5 units up.
Meaning, C = 7.5

Therefore, y = (1.5)cos[(pi/5)x] +7.5 ---------------answer.

Check for max y. It s at (0,9):
y = (1.5)cos[(pi/5)*0] +7.5
y = (1.5)cos(0) +7.5
y = 1.5(1) +7.5
y = 9
So, (0,9) ----checks

For min, at (5,6):
y = (1.5)cos[(pi/5)(5)] +7.5
y = (1.5)cos(pi) +7.5
y = 1.5(-1) +7.5
y = 6
So, (5,6) ----checks again.