Does the marginal product need to be diminishing to have a diminishing average product? We have the rule 'If the marginal is less than the average, then the average declines'.

So if our production function for an input x and output y is concave, does that mean that MP must be diminishing given AP is diminishing?
Suppose the total product is given by $f(x)$. Thus, the average product is $$AP(x)=\frac{f(x)}{x}.$$

Differentiating:
$$AP'(x)=\frac{x f'(x)-f(x)}{x^2},$$

which has the same sign as $x f'(x)-f(x)$. This is definitely negative if $f''(x)&lt;0$ (i.e., if $f$ is concave).

But it can also be negative, even if $f$ is locally convex. Take a look at this figure:



At the point marked with a blue disc: the slope of the red line is $f'(x)$. The slope of the black line is $f(x)/x$. We can see geometrically that $f(x)/x&gt;f'(x)$, or (after rearranging) that $f'(x)x&lt;f(x)$.

Thus, at the point indicated by the blue disc we know that $AP'(x)&lt;0$. But at this point we can also see in the figure that $f''(x)&gt;0,$ implying that the marginal product is increasing.

Since we have a counter-example, we conclude that a decreasing marginal product is not a necessary condition for a decreasing average product.