A company ships $x$ units per order. Its annual cost is $C(x)=\frac{4000}{x}+10x+250$ dollars for $x>0$. What order size minimizes cost?

A supplier's annual ordering cost is $C(x)=\frac{7200}{x}+8x+300$ dollars for $x>0$, where $x$ is units per order. What order size minimizes cost?

A warehouse's annual cost is $C(x)=\frac{1800}{x}+5x+90$ dollars for $x>0$. In exact form, what order size minimizes cost?

A factory's monthly cost is $C(x)=0.02x^2-8x+1400$ dollars for $0\le x\le 400$ units. What is the minimum monthly cost?

A distributor's annual cost is $C(x)=\frac{12000}{x}+3x+400$ dollars for $x>0$. To the nearest tenth, what order size minimizes cost?

A concert sells $q=600-15p$ tickets when the ticket price is $p$ dollars. What ticket price maximizes revenue?

A shop's price-demand equation is $p=50-0.5q$, where $p$ is dollars per package. What is the maximum daily revenue?

A theater sells $q=900-30p$ tickets at price $p$ dollars. Its cost is $6q+900$ dollars. What ticket price maximizes profit?

A bakery sells $q$ loaves daily at price $p=30-0.1q$ dollars and has cost $C(q)=4q+200$ dollars. What is its maximum daily profit?

A museum sells $q=500-10p$ admissions when the price is $p$ dollars. What admission price maximizes revenue?

A runner's travel time is $T(x)=\frac{100-x}{5}+\frac{\sqrt{x^2+1600}}{2}$ seconds for $0\le x\le 100$. Which equation locates the time-minimizing value of $x$?

A swimmer runs $x$ meters along shore at 4 meters per second, then swims straight at 2 meters per second to a buoy 30 meters offshore. The buoy is opposite a point 80 meters along shore from the start. What $x$ minimizes time?

A hiker crosses a 30-meter-wide field diagonally at 3 meters per second to a point $x$ meters along the far edge, then walks the remaining distance to a point 120 meters away at 6 meters per second. What $x$ minimizes time?

The concentration of a drug is $C(t)=\frac{4t}{t^2+1}$ milligrams per liter for $t\ge 0$. At what time is the concentration greatest?

A tank's capacity $V$ is a continuous function of $x$ on the closed interval $[0,25]$. Which theorem guarantees that $V$ actually attains an absolute maximum somewhere on $[0,25]$, before any candidate values are even computed?

A 30-centimeter-wide metal strip is bent upward equally on both sides to form a one-meter trough. Its capacity is $V(x)=\frac{x(30-2x)}{10}$ liters, where $x$ is in centimeters. What value of $x$ maximizes capacity?

A chemical concentration is modeled by $C(t)=\frac{6t}{t^2+9}$ milligrams per liter for $t\ge 0$. At what time is the concentration greatest?

A cost function is continuous on $[0,30]$ and differentiable on $(0,30)$. Its only critical point in $(0,30)$ is $x=12$. Values are $C(0)=920$, $C(12)=700$, and $C(30)=800$. What establishes the absolute minimum?

A differentiable cost function on $x>0$ has one critical point at $x=20$. Which statement proves it is an absolute minimum there?

Revenue is $R(p)=-20p^2+800p$ for $0\le p\le 40$. A critical point occurs at $p=20$. What justifies that this price gives the absolute maximum revenue?