Minimum area between f(x) and a tangent line

[brb8705]

How would you write a proof that proves that the minimum area between a function and its tangent line is the tangent line evaluated at point p, where p is the midpoint on a given interval?

i.e. The minimum area between x^2, and its tangent line on the interval [0,1] is the tangent line evaluated at x=1/2

Thanks,

 

[Office_Shredder]

I don't think that's true

 

[HallsofIvy]

Which tangent line are you talking about? A function may have an infinite number of distinct tangent lines on an interval.

 

[brb8705]

Right, out of all the tangent lines of a function on a interval which one is the minimum area between the function and a tangent line.

I'm looking for a proof to prove this:
The minimum area between x^2, and its tangent line on the interval [0,1] is the tangent line evaluated at x=1/2

 

[Wizlem]

Just find the area as a function of p and then differentiatie and find where dA(p)/dp=0 where A is the area to find the extrema. Figure out if any is a minima and then also check endpoints of your range.