17. Incorrect. The answer is true, not false

(a) The mode corresponds to the point where the density f(z) is a maximum. The relative maximum of f(z) occur where the derivative of the density function is zero:

df/dz[4z(9-z²)/81] = (36-12z²)/81 =0.

Then z* = 3^0.5 ≈ 1.73 (US$ 1730) is the mode. Check that the second derivative, -24z/81 is negative at z* = 3^1/2. So, z* defines a maximum.

(b) The median is the value at which P(Z≤c) = 1/2. For 0<c<3:

P(Z≤c) = 4/ 81 ∫₀^c z(9-z ²)dz

P(Z≤c) = 4/ 81(9c²/2 – c⁴/40

Setting this equal to 1/2:

2c⁴-36c²+81=0

c² = 9(9/2)*2^1/2

Since c must be between 0 and 3, c² = 9- (9/2)*2^1/2

c ≈ 1.62 (US$ 1620).

(c) E(Z) =4/81 ∫₀³ z² (9-z ²)dz

E(Z) = 4/81 (3z – z⁵ / 5│₀³

E(Z) = 1.60 (US$ 1600).

Mean < Median < Mode.