The supply function of a commodity is $P = x^3+2x+18$. When 4 units of commodity are sold, then producer surplus is:

208

The correct answer is Option (4) → 208 **

Supply function: $P = x^{3} + 2x + 18$

Producer surplus at $x=4$:

$PS = xP(x) - \displaystyle \int_{0}^{x} P(t)\,dt$

Compute $P(4)$:

$P(4)=4^{3}+2\cdot 4+18=64+8+18=90$

Revenue at 4 units:

$4 \times 90 = 360$

Compute the integral:

$\displaystyle \int_{0}^{4}(t^{3}+2t+18)\,dt$

$= \left[\frac{t^{4}}{4} + t^{2} + 18t\right]_{0}^{4}$

Substitute $4$:

$\frac{4^{4}}{4}+4^{2}+18\cdot 4 = 64 + 16 + 72 = 152$

Producer surplus:

$PS = 360 - 152 = 208$

Producer surplus = 208