# Differentiation Question

1. Jan 15, 2010

### GunnaSix

1. The problem statement, all variables and given/known data
Find a function of the form $$f(x) = a + b \cos cx$$ that is tangent to the line $$y = 1$$ at the point $$(0,1)$$, and tangent to the line $$y = x + 3/2 - \pi /4$$ at the point $$(\pi /4 , 3/2)$$.

2. Relevant equations

3. The attempt at a solution
$$f(0) = a + b = 1$$, so $$a = 1 - b$$.

This is as far as I can get though.

$$f'(0) = -bc \sin cx = 0$$

for any a, b, and c, and

$$f(\pi /4) = (1 - b) + b \cos [(\pi /4)c] = 3/2$$

and

$$f'(\pi /4) = -bc \sin [(\pi /4)c] = 1$$

don't really seem to help me.

What am I missing?

2. Jan 16, 2010

### snipez90

Well you can combine the last two equations to get
$$b\left[\cos\left(\frac{c\pi}{4}\right)-c\sin\left(\frac{c\pi}{4}\right)-1\right] = \frac{3}{2}.$$
Presumably this will give you infinitely many solutions. For instance, c = 2 works.

3. Jan 16, 2010

### payumooli

the function has a maximum at x = 0, because ymax = a+b
if you do the second derivative test you will find -bc^2 < 0
so b>0

i am not able to tell more than this from the given data