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Explanation:

The cost function C(x) = 240 + bx + cx2
C(20) = 240 + 20b + 400c
C(40) = 240 + 40b + 1600c
C(60) = 240 + 60b + 3600c

By conditions,

2/3 × C(20) = C(40) – C(20)
⇒ C(40) = 5/3 × C(20)
⇒ 240 + 40b + 1600c = 400 + 100b/3 + 2000c/3
⇒ 20b/3 + 2800c/3 = 160
⇒ 20b + 2800c = 480 ... (1)

Also,

½ × C(40) = C(60) – C(40)
⇒ 3/2 × C(40) = C(60)
⇒ 360 + 60b + 2400c = 240 + 60b + 3600c
⇒ c = 1/10
⇒ b = 10 ... (from 1)

Profit P(x) for x units is 30x – C(x)
⇒ P(x) = 30x  – 240 – 10x – x2/10 = – x2/10 +20x – 240

This is a quadratic expression whose maximum value will occur at x = -(20/2 × -1/10) = 100
∴ x = 100

Hence, option (b).

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