If f (x) = x ^ 3-3x ^ 2-3mx + 4 has a maximum value of 5, find the value of the real number m and the tangent equation of the curve y = f (x) passing through the origin

If f (x) = x ^ 3-3x ^ 2-3mx + 4 has a maximum value of 5, find the value of the real number m and the tangent equation of the curve y = f (x) passing through the origin

Solution f (x) '= 3x ^ 2-6x-3m
Let f (x) '= 3x ^ 2-6x-3m = 3 (x-1) ^ 2-3m-3 = 0,
It is reduced to (x-1) ^ 2 = m + 1, which must have a solution, so m > = - 1, the maximum of the secondary equation is at the first inflection point, and
X = - (M + 1) ^ 1 / 2 + 1, the value obtained by substituting this x value into the original equation is 5, the solution is m = - 1, and the extreme point is x = 1,
F (0) '= 3x ^ 2-6x + 3 = 3, the tangent equation through the origin is y = 3x