Characteristics of Average & Marginal Product

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Here, x₂is fixed. H is the point of inflexion. Ap = 0 means TP = 0.

TP is highest when MP = 0.

Characteristics:
1. AP for a point on TPC (TP curve) equals to the slope of a line segment connecting that point with the origin.
2. AP increases for movements along the TPC from the origin to point J and decreases thereafter. Point J corresponds to the maximum point on the AP curve.
3. MP for a point on TPC equals the slope of the tangent to the curve at that point. MP increases from the origin to the point of inflexion H where the slope of the tangent is at a maximum and decreases thereafter.
4. MP and AP are equal at the maximum of AP.

Prove AP equals MP when AP is maximum:
Proof:
To determine the maximum value of AP, its partial derivative with respect to x₁ equals to zero.
i.e.
δAP/δx1 = 0
We know, 
AP = f (x₁, x₂^o)/x₁
∴ Differentiating w.r.t x₁ ,
{x₁f₁ (x₁, x₂^o) - f (x₁, x₂^o) }/x₁² = 0
=> x₁f₁ (x₁, x₂^o) - f (x₁, x₂^o)  = 0
=> f₁ (x₁, x₂^o) = { f (x₁, x₂^o)}/ x₁ 
=> AP = MP