Why must the marginal cost rise at the point of Producer's Equilibrium?

The marginal cost must be rising to attain the producer's equilibrium because the producer's equilibrium is attained through the profit-maximizing process.

If marginal cost is falling, then the producer can always increase its profits by producing more output. The reason is that producing an additional production will cost less to the producer.

Mathematically, total profits are calculated as total revenue minus total costs. When we maximize total profits, we get marginal cost equals marginal revenue as our first order derivative condition. The second-order derivative condition will show the marginal cost to be rising. Therefore, we prove that marginal cost should be rising at the point of the producer's equilibrium.