Let f (x) have the second derivative, try to prove that the abscissa a of the inflection point of the curve y ^ 2 = f (x) is suitable for the following relation (f '(a)) ^ 2 = 2F (a) f' '(a)

Let f (x) have the second derivative, try to prove that the abscissa a of the inflection point of the curve y ^ 2 = f (x) is suitable for the following relation (f '(a)) ^ 2 = 2F (a) f' '(a)

Prove: y ^ 2 = f (x), so: y = f (x) ^ 1 / 2Y '= 1 / 2F (x) ^ (- 1 / 2) * f' (x) inflection point, y '' = 1 / 2 * (- 1 / 2) f (x) ^ (- 3 / 2) * f '(x) * f' (x) + 1 / 2F (x) ^ (- 1 / 2) * f '' (x) = 0, the abscissa of inflection point is a, so 1 / 2F (x) ^ (- 1 / 2) * f '' (a) = 1 / 4f (x) ^ (- 3 / 2) * [f '(a)] ^ 2F' '(a) = 1 / 2F (x) ^ (