b
(b)

\(T=-\alpha V^3+\beta V^2 \quad \ldots (i)\)

and \(P V=n R T \quad... (ii)\)

\(n=1\)

So, \(P=\frac{R T}{V}\)

Multiplying \(\frac{R}{V}\) in \((i)\)

\(\frac{R T}{V}=\left(-\alpha V^2+\beta V\right) R\)

or \(P=\left(-\alpha V^2+\beta V\right) R \ldots (iii)\)

\(\frac{d P}{d V}=(-2 \alpha V+\beta) R\)

Maxima is when \(\frac{d P}{d V}=0\) and \(\frac{d^2 P}{d V^2}\) in negative, so

\(O=(-2 \alpha V+\beta) R\)

\(V=\frac{\beta}{2 \alpha}\)

Put in value of \(V\) in equation \((iii)\)

\(P=\left(-\alpha \frac{\beta^2}{4 \alpha^2}+\frac{\beta^2}{2 \alpha}\right) R \Rightarrow P=\frac{\beta^2 R}{4 \alpha}\)