= Solution
{c}
The stated weighted <inequality> is the <Jensen inequality> for a <concave function>. Assume $a<b$; if $a=b$, both sides of the required <inequality> are zero. Write $x=\lambda a+(1-\lambda)b$ with $\lambda=(b-x)/(b-a)$. Applying the weighted <inequality> at the two endpoints gives
$$
f(x)\geq\ell(x)=\frac{b-x}{b-a}f(a)+\frac{x-a}{b-a}f(b).
$$
Thus the <affine function> $\ell$ joining the endpoint values lies below $f$. By <monotonicity of the Riemann integral>,
$$
\int_a^b f(x)\,dx\geq\int_a^b\ell(x)\,dx
=\boxed{\frac{b-a}{2}\bigl(f(a)+f(b)\bigr).}
$$
This is the lower bound in the <Hermite-Hadamard inequality> for a <concave function>.
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