Expert: Paul Klarreich - 4/13/2007

Question

"this problem is a multi choice one: if f'(x) and g'(x)exist and f''(x)> g'(x)
for all real x, then the graph og y=f(x) and the graph of y=g(x)
a). intersect exactly once
b). intersect no more than once
c). do not intersect
d). could intersect more than once
e). have a common tangent at each point of intersection
thanks"


Answer
Questioner:   tina
Category:  Calculus
 
Subject:  calculus
Question:  "this problem is a multi choice one: if f'(x) and g'(x)exist and f''(x)> g'(x) for all real x, then the graph of y=f(x) and the graph of y=g(x) a). intersect exactly once b). intersect no more than once c). do not intersect d). could intersect more than once e). have a common tangent at each point of intersection thanks"
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Hi, Tina,

I think you mean that  f' > g' for all x.  In that case, let   h(x) = f(x) - g(x)

If h(x) = 0 anywhere, then f and g intersect, so consider that.

h'(x) = f'(x) - g'(x) and so

h'(x) > 0 for all x

Now what if the graphs of f and g intersect in two points,  a and b?

Then h(a) = 0 and h(b) = 0.  According to the MVT, there must be a value of c in the open interval (a,b) where  h'(c) = 0.  But  h'(x) > 0 for all x, therefore no such c can exist and therefore the graphs of f and g can intersect at most once.

Must they intersect at all?  No.  Here is an example:

Let f(x) = 2e^x, then f'(x) = 2e^x
Let g(x) = e^x,  then g'(x) = e^x

Now  f'(x) - g'(x) = 2e^x - e^x = e^x > 0, so these functions satisfy the conditions.

Do they intersect?  Set  f(x) = g(x).  


Solve 2e^x = e^x.  This gives  e^x = 0, which is impossible since e^x > 0 for all x.