= Solution
With $q=p'$ and $r=0$, the equation is
$$
y''''+(py')'=\lambda y.
$$
Multiply by the real <eigenfunction> $y$ and integrate over $[a,b]$. The <clamped boundary condition>[clamped boundary conditions] $y=y'=0$ at both endpoints eliminate all boundary terms, so
$$
\lambda\int_a^b y^2dx
=\int_a^b(y'')^2dx-\int_a^b p(x)(y')^2dx.
$$
Since $p(x)<0$, the right side is a sum of nonnegative terms. It cannot vanish for a nonzero eigenfunction: if $y''=y'=0$, the boundary conditions force $y=0$. Thus
$$
\boxed{\lambda
=\frac{\int_a^b\{(y'')^2-p(y')^2\}\,dx}{\int_a^b y^2dx}>0}.
$$
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