The function g (x) = - 2lnx + ax - (3a + 2) / X is not monotone in the interval [1,4], and the range of a is obtained Using derivative to do classification discussion

The function g (x) = - 2lnx + ax - (3a + 2) / X is not monotone in the interval [1,4], and the range of a is obtained Using derivative to do classification discussion

For this kind of problem, sometimes it is difficult to solve from the front, then we might as well change a way of thinking, assuming that the function is monotone on [1,4], find out the value range of a, and finally find out the correct solution
To derive a function,
g'（X）=-2/x+x+(3a+2)/x*x
Because the function is not monotone in the interval [1,4], there must be a point X in the interval [1,4], so that G '(x) = 0
Let g '(x) = 0, - 2 / x + X + (3a + 2) / X * x = 0, - 2 * x + X * x + 3A + 2 = 0
Let f (x) = - 2 * x + X * x * x + 3A + 2, f '(x) = 3 * x * x - 2, f (x) monotonically increases on [1,4], and the value range of F (x) [3A + 1,58 + 3A]
3a+10,-58/3