Question

Determine the points of maxima and minima of the function f(x) = (1/8)(ln x - bx + x²), x > 0 where b ≥ 0 is a constant.

Answer 1

f(x) is a differentiable function for x > 0. Therefore, for maxima or minima f'(x) = 0 must satisfy.

Case I : 0 ≤ b <1, (i) has no solution.

Since RHS is negative in this domain and LHS is positive.

Case II ; b = 1,

x = 1/4 is the only solution. When b = 1

From sign chart, it is clear that f(x) has no change of sign in left and right of x = 1/4.