Quantum Mechanics problem

Homework Statement

Apply direct differentiation to the ground state wave function for the harmonic oscillator,

Psi=A*e^(-sqrt(mk)x^2/(2*h_bar))*e^(-i*w*t/2) (unnormalized)

and show that Psi has points of inflection at the extreme positions of the particle's classical motion.

The Attempt at a Solution

My understanding of this question is first I have to normalize the wave function to get the value of the constant A. Then I must differentiate twice and set the second derivative of the wave function to zero and solve the resulting equation. When I do that I get an equation whose solution is complex and is different than that required by the question.

Normalization:

after normalization I get A=(m*k)^1/8 / (2*(pi*h_bar)^1/4)

Differentiation:

d(Psi)/dx=A*(-sqrt(m*k) / h_bar) e^(-i*w*t/2) * x *e^(-sqrt(m*k)x^2/(2*h_bar))

d^2(Psi)/dx=A * e^(-i*w*t/2) * [(-sqrt(m*k) / h_bar) * e^(-sqrt(m*k)x^2/(2*h_bar)) + 4 * x^2 * (-sqrt(m*k) / 2*h_bar)^2 * e^(-sqrt(m*k)x^2/(2*h_bar))]

Setting d^2(Psi)/dx=0 I get:

(1+0.5*x^2) * e^(-sqrt(mk)x^2/(2*h_bar)) = 0

Whose solution is: 1.414213562 i, -1.414213562 i