|Nov6-10, 02:30 AM||#1|
Construct a piecewise function f(x)
1. The problem statement, all variables and given/known data
Construct a piecewise function f(x) such that the following conditions are satisfied:
Concave down on ( - infinity, 5)
Concave up on (5, infinity)
increasing over ( -infinity,infinity)
Nonzero- sloped tangent line at the point of inflection x = 5
f(5) = 11 (continuous at x = 5)
f ' (5) =6
3. The attempt at a solution
I really don't know what I should do to get started. My teacher assigned this as a homework problem without going over anything related to this. I could draw a graph to easily satisfy the given conditions, but I don't know how to find the equations of the curves I drew.If you could point me in the right direction I would really appreciate it.
|Nov6-10, 04:37 AM||#2|
The condition f(5) = 11 is not a problem. Geometrically, we can shift any graph we find up or down by the right amount to fulfill this. In equations, this is just adding a constant to the end of whatever piecewise function we define. (Why?)
The condition of non-zero sloped tangent line at x = 5 is redundant, because f'(5) = 6.
What is an increasing concave up function? What is an increasing concave down function? Even if you cannot find one that is increasing everywhere, can you find one that is increasing over part of its domain? You can perhaps chop it off where you need it.
In particular, if your two candidate functions off of which you will take pieces are f(x) and g(x). Find when f'(x) = 6 and g'(x) = 6. Shift the graph left or right (how do you do this?) until both have derivative 6 at x = 5. Shift the graph up or down until both are equal to 11 at x = 5.
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