4.4: Optimization I

Absolute Extrema

If $f(x) \leq f(c)$ for all $x$ in the domain of $f$, then $f(c)$ is called the absolute maximum value of $f$.
If $f(x) \geq f(c)$ for all $x$ in the domain of $f$, then $f(c)$ is called the absolute minimum value of $f$.
Absolute extrema might not exist.
If a function $f$ is continuous on a closed interval $[a,b]$ then $f$ has both an absolute maximum value and an absolute minimum value on $[a, b]$.
To find absolute extrema, just find critical numbers and check endpoints of the interval.
Finding Absolute Extrema on a Closed Interval
  1. Find the critical number of $f$ in $(a, b)$.
  2. Keep track of the value of $f$ at each critical number and compute $f(a)$ and $f(b)$.
  3. The absolute maximum and absolute minimum value will correspond to the smalles and largest values found in step 2.
Find the absolute extrema of $f(x) = x^2$ on $[-1, 2]$.
Find the absolute extrema of the function \[f(x) = x^3 - 2x^2 - 4x + 4\] on $[0, 3]$.
Find absolute extrema of \[f(x) = x^{2/3}\] on $[-1, 8]$.
Acrosonic's total profit in dollars from manufacturing and selling $x$ units of their loudspeaker is given by \[P(x) = -0.02x^2 + 300x - 200000 \qquad 0 \leq x \leq 20000\] How many units of the loudspeaker system must Acrosonic produce to maximize its profits?
The daily average cost function (dollars per unit) of some company is given by \[\bar{C}(x) = 0.0001x^2 - 0.08x + 40 + \dfrac{5000}{x} \qquad x \geq 0\] Show that a production level of 500 units per day results in a minimum average cost for the company.
The altitude (in feet) of a rocket $t$ seconds into flight is given by \[f(t) = -t^3 + 96t^2 + 195t + 5 \qquad t \geq 0\] Find the maximum altitude and maximum velocity attained by the rocket.