Show that the next eigenfunction has a zero between 2 zeros

[Happiness]

Homework Statement

Homework Equations

Wronskian theorem:

The Attempt at a Solution

I've gotten the relationship given by the question but I do not know how to continue.

Since $\psi_n(a)=\psi_n(b)=0$,
LHS $=\psi_n'(b)\,\psi_{n+1}(b)-\psi_n'(a)\,\psi_{n+1}(a)$

If LHS $=0$, RHS $=0$, then by the First Mean Value Theorem for Integrals, $\psi_{n+1}(c)=0$ for some $c\in(a,b)$.

But LHS is not necessarily $0$.

Attached below is the derivation of Wronskian theorem.

 

[Samy_A]

As a and b are consecutive zeros of $\psi_n$, $\psi_n$ won't change sign between a and b.
Assume, without loss of generality, that $\psi_n$ is positive between a and b.
What does that tell you about the signs of $\psi_n'(a)$ and $\psi_n'(b)$?

Now assume that $\psi_{n+1}$ has no zeros between a and b. It follows that $\psi_{n+1}$ won't change sign between a and b.
Now just check what the signs of the two expressions will be (remember that $E_{n+1}>E_n$).