Question

The cost C for producing x articles is given as C = x³ − 16x² + 47x. For what values of x, the average cost is decreasing?

Solution:

Given C = x³ − 16x² + 47x

Average cost `C_A = C/x`

∴ `C_A = square`

Differentiating w.r.t. x, we get

`d/dx (C_A) = square`

We know that C_A is decreasing, if

`d/dx (C_A) square 0`

∴ 2x − 16 < 0

∴ 2x < 16

∴ x < `square`

∴ average cost is decreasing for x ∈ (0, 8).

Answer 1

Given C = x³ − 16x² + 47x

Average cost  \[C_A=\frac{C}{x}=\frac{x^3-16x^2+47x}{x}\]

∴ C_A = \[\boxed{x^2 - 16x + 47}\] \[\]

Differentiating w.r.t. x, we get

\[\frac{d}{dx}(C_A)=\frac{d}{dx}(x^2-16x+47)\]

= 2x − 16 × 1 + 0

= \[\boxed{2x - 16}\]

We know that C_A is decreasing, if

\[\frac{d}{dx}(C_A)\boxed{<}\] 0

∴ 2x − 16 < 0

∴ 2x < 16

∴ x < \[\boxed{8}\]

∴ average cost is decreasing for x ∈ (0, 8).