单变量优化

The flat-tangent principle

Hilltop or valley floor

Marginal reasoning as the FOC

Boundaries deserve a glance

Why does an interior maximum of a smooth function satisfy $f'(x) = 0$?

A smooth function satisfies $g'(x) = 0$ at $x = 7$ and $g''(x) = 3$ everywhere. What is at $x = 7$?

Profit is $\pi(q) = 40q - q^2 - 100$. Which pair gives the optimal output and the maximum profit?

Profit is $\pi(q) = 10q - q^2$ and capacity limits output to at most 4 units. Where does the firm produce?

For $\pi(q) = 40q - q^2 - 100$, solve the first-order condition for the optimal output $q^*$, and write the number.

Compute the maximum profit $\pi(20)$ for the same function, and write the number.

For profit 10q minus q squared with a capacity ceiling of 4, what is the best feasible output?

In the decomposition with revenue 40q minus q squared over 2, what is marginal revenue?

At the optimal output of 20, what do marginal revenue and marginal cost each equal?

What does the second derivative minus 2 guarantee about the first-order condition's answer?

Compute profit at q of 4 under 10q minus q squared, that is 40 minus 16, and write the number.

Differentiate cost q squared over 2 plus 100 and evaluate the margin at q of 6. Write the number.