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The function f(x), defined for x>=0 has the following property: For any real number a>=0, the area between the graph of y=f(x) for 0<=x<=a and the graph of y=sqrt(x) is equal to the area between the same part of the graph and the y-axis. (In other words, the red area is equal to the blue area.)

(a) Find a differential equation that the function f(x) satisfies. (In particular, this equation will involve f(x) and f'(x).)
(b) Prove that f(x) = 2sqrt(x) + kx for some constant k. 
 

 Jul 14, 2021
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Try using a Taylor series expansion.

 Jul 14, 2021

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