Calculus - finding the minimal length between two functions

[Femme_physics]

I'll have to do some backtracking to see how this all makes sense. I lost the string of the actual problem with all the math mishaps. Thank you all though! Much appreciate ILS for going for the kill! :)

 

[Mentallic]

I'll have to do some backtracking to see how this all makes sense. I lost the string of the actual problem with all the math mishaps. Thank you all though! Much appreciate ILS for going for the kill! :)

The basic idea of how to answer this question - without all the maths - is we first start with some function f(x) and another g(x), these functions don't intersect by looking at their graphs (we could also have proven this by setting f(x)=g(x) and shown there are no real solutions too, but it wasn't necessary here) and now to find the minimal distance in the y-direction between each function, we consider that the function f(x)-g(x) describes their differences in their height y at each point x, so for example if we took the value x=3, then f(3)=1/4 and g(3)=-4/3, which means their height difference is f(2)-g(2)=1/4-(-4/3)=19/12. So the function f(x)-g(x) is their height difference, and finding the minimal height means finding the lowest point on this graph (it will be above the y-axis, because we've already argued that the graphs don't intersect so f(x)-g(x)$\neq$0) and this is where calculus comes into it. Since when we find the derivative of a function, it tells us the gradient of the tangent at each x value, well we know the lowest point on the function has a tangent gradient of 0, so we find the derivative and set it to zero (this says "tangent gradient = 0") then solving for x, it will tell us the x-value where lowest point is. Now substituting x back into the derivative equation obviously yields 0, because we just solved the equality and found the value of x that makes the derivative 0. We need to substitute the x-value back into the function f(x)-g(x) which will give us the y-value, which is the height difference between the functions. This is it.