Transcript Global max/min [4.3]

Section 4.3
Global Maxima and Minima
Applied Calculus ,4/E, Deborah Hughes-Hallett
Copyright 2010 by John Wiley and Sons, All Rights Reserved
Figure 4.30: Global maximum and minimum
on an interval domain, a ≤ x ≤ b
Figure 4.31: Global maximum and minimum
on the entire real line
Applied Calculus ,4/E, Deborah Hughes-Hallett
Copyright 2010 by John Wiley and Sons, All Rights Reserved
Consider the graph
of f(x) in the figure.
Where is the global maximum of f(x)?
𝑥5
Where is a local maximum that is not a global maximum of f(x)?
𝑥2 , 𝑥8 , 𝑥10
Where is the global minimum of f(x)?
𝑥3 and 𝑥9
Find the global maximum and minimum for 𝑓 𝑥 = 3 + 5𝑥 − 𝑥 2 on
the interval [1, 5].
The global maximum is at 𝑥 = 2.5
The global minimum is at 𝑥 = 5.
Find the global maximum and minimum for 𝑓 𝑥 = 𝑥 − ln 𝑥 on the
interval (0, ∞).
There is no global maximum.
The global minimum is at 𝑥 = 1.
On the same side of a straight river are two towns, and the
townspeople want to build a pumping station, S. See Figure 4.9. The
pumping station is to be at the river’s edge with pipes extending
straight to the two towns. Where should the pumping station be
located so as to minimize the total length of the pipe?
We need to minimize
d( x)  x 2  1  (4  x)2  16
The global minimum is at 𝑥 = 4/5.
The table records the rate of change of air temperature, H, as a function
of time, t, one morning.
When is it coldest?
11:00AM
What temperature was it at that time?
Cannot tell from this information
If it was 40 ◦F at 7:00AM, estimate the temperature at 11:00AM.
40 − 3 − 0.5 − 1 − 3 = 32.5 ◦F
When might there have been an inflection point in H?
sometime between 10:00AM and 11:00AM
t (hours since midnight)
dH/dt (◦F/hour)
7
−5
8
−1
9
0
10
−2
11
−4
Following a dose a of a drug, the average blood pressure of a patient is b.
To compute the dose that maximizes the blood
pressure, we
(a) Find da/db and substitute b = 0.
(b) Find db/da and substitute b = 0.
(c) Solve for the values of a for which da/db = 0.
(d) Solve for the values of a for which db/da = 0.
Problem 16
Figure 4.38 shows the rate at which photosynthesis is taking place in a leaf.
(a) At what time, approximately, is photosynthesis proceeding
fastest for t ≥ 0?
50 days
(b) If the leaf grows at a rate proportional to the rate of
photosynthesis, for what part of the interval 0 ≤ t ≤ 200
is the leaf growing? When is it growing fastest?
t > 0 days; 50 days
Figure 4.38
Applied Calculus ,4/E, Deborah Hughes-Hallett
Copyright 2010 by John Wiley and Sons, All Rights Reserved