Marginal Product and Average product So I've seen in my textbooks and many places online that when marginal product is more than average product average product is increasing, when average product is falling average product is greater than the marginal product. I get the economic intuition behind all this but I was wondering how can one show this mathematically.
Here is a simple explanation for a single input and output technology represented by a production function $f:\mathbb{R} \to \mathbb{R}$.

Note that Average Product (AP) is defined as 
$$
AP(x) = \frac{f(x)}{x}
$$
, and Marginal Product (MP) is defined as
$$
MP(x) = \frac{\partial f(x)}{\partial x} = f'(x)
$$

Now, we take the derivative of the function $AP(x)$, and then we have
$$
\frac{\partial AP(x)}{\partial x} = \frac{f'(x)x-f(x)}{x^2}
\\
= \frac{1}{x}\bigg[f'(x)-\frac{f(x)}{x}\bigg]
\\
= \frac{1}{x}[MP(x)-AP(x)]
$$

Since $x$ indicates the level of the input, $x&gt;0$, and thus we have,
$$
\frac{\partial AP(x)}{\partial x} &gt; 0 \iff MP(x)&gt;AP(x)
$$
, or 
$$
\frac{\partial AP(x)}{\partial x} &lt; 0 \iff MP(x)&lt;AP(x)
$$