Thread: One more Calculus Problem

4. Let f(x) and g(x) be the differentiable functions graphed here. At x=c, the vertical distance between these curves is the greatest. Is there anything special about the tangents to the two curves at x=c? Give reasons for your answers. f(x) is the line that is concave up, and g(x) is the line that is concave down.

I am in the first semester of calculus, so this would have to apply to that.... but I have no clue where to start on this one :P. So help would be nice.

Forgot to put the graph here.

Originally Posted by Prophet

4. Let f(x) and g(x) be the differentiable functions graphed here. At x=c, the vertical distance between these curves is the greatest. Is there anything special about the tangents to the two curves at x=c? Give reasons for your answers. f(x) is the line that is concave up, and g(x) is the line that is concave down.

I am in the first semester of calculus, so this would have to apply to that.... but I have no clue where to start on this one :P. So help would be nice.

the vertical distance between the functions at any value of is given by . define a new function, , to be this distance. thus we have:



now we want to maximize . so we find its derivative and set it equal to zero, thus we have:



what can we say about and ? which are the slopes of the tangent lines of and respectively