Section 4.1
Local Maxima and Minima
Applied Calculus ,4/E, Deborah Hughes-Hallett
Copyright 2010 by John Wiley and Sons, All Rights Reserved
y
How could the local
maxima and minima
have been found?
Figure 4.2: Graph of f (x) = x3 – 9 x2 – 48 x +52
Applied Calculus ,4/E, Deborah Hughes-Hallett
Copyright 2010 by John Wiley and Sons, All Rights Reserved
Figure 4.6: Changes in direction at a critical point, p: Local maxima or minima
Applied Calculus ,4/E, Deborah Hughes-Hallett
Copyright 2010 by John Wiley and Sons, All Rights Reserved
Consider the graph of f(x).
What are the critical points of 𝑓?
𝑥2 , 𝑥5 , 𝑥6 , 𝑥7 , 𝑥8 , 𝑥9
What are the local minima of 𝑓?
𝑥7 , 𝑥9
What are the local maxima of 𝑓?
𝑥2 , 𝑥5 , 𝑥8
How do the above answers change if
the upper hole at 𝑥3 is filled in?
𝑥3 becomes a critical point
How do the above answers change if
the lower hole at 𝑥3 is filled in?
𝑥3 becomes a critical point
and a local minimum
Find all critical points, local minima, and local maxima of the following
functions.
𝑓 𝑥 = 4𝑥 3 + 3𝑥 2 − 36𝑥 − 5
3
2
𝑥 = −2 is a local maximum and 𝑥 = is a local minimum
g 𝑥 = 𝑥 − 2 ln(𝑥 2 + 3)
𝑥 = 1 is a local maximum and 𝑥 = 3 is a local minimum
The table records the rate of change of air temperature, H, as a function
of time, t, during one morning. When was the temperature a local
minimum? Local maximum?
t (hours since midnight)
dH/dt (◦F/hour)
6
1
A local maximum occurs at 8:00AM
A local minimum occurs at 10:00AM.
7
2
8
0
9
−2
10
0
11
3
12
2
Which of the following pieces of information from a daily
weather report allow you to conclude with certainty that there
was a local maximum of temperature at some time after 10:00
am and before 2:00 pm?
(a) Temperature 50◦ at 10:00 am and 50◦ and falling at 2:00 pm.
Yes
(b) Temperature 50◦ at 10:00 am and 40◦ at 2:00 pm.
No, may have been strictly decreasing.
(c) Temperature rising at 10:00 am and falling at 2:00 pm.
Yes
(d) Temperature 50◦ at 10:00 am and 2:00 pm, 60◦ at noon.
Yes
(e) Temperature 50◦ at 10:00 am and 60◦ at 2:00 pm.
No, may have been strictly increasing.
If the graph in Figure 4.2 is that of f′(x), which of the following
statements is true concerning the function f?
(a) The derivative is zero at two values of x, both being local
maxima.
(b) The derivative is zero at two values of x, one is a local maximum
while the other is a local minimum.
(c) The derivative is zero at two values of x, one is a local maximum
on the interval while the other is neither a local maximum nor a
minimum.
(d) The derivative is zero at two values
of x, one is a local minimum on the
interval while the other is neither
a local maximum nor a minimum.
(e) The derivative is zero only at one
value of x where it is a local minimum.
Figure 4.2
Problem 7
Graph two continuous functions f and g, each of which has exactly
five critical points, the points A-E in Figure 4.12, and which satisfy
the following conditions:
(a) f (x) → ∞ as x → – ∞ and
f (x) → ∞ as x → ∞
(b) g (x) → – ∞ as x → – ∞ and
g (x) → 0 as x → ∞
Figure 4.12
Applied Calculus ,4/E, Deborah Hughes-Hallett
Copyright 2010 by John Wiley and Sons, All Rights Reserved
Problem 36
Assume f has a derivative everywhere and just one critical point,
at x = 3. In parts (a) – (d), you are given additional conditions. In
each case, decide whether x = 3 is a local maximum, a local
minimum, or neither. Sketch possible graphs for all four cases.
(a) f ‘(1) = 3 and f ‘(5) = – 1
(b) f (x) → ∞ as x → ∞ and as x → – ∞
(c) f (1) = 1, f (2) = 2, f (4) = 4, f (5) = 5
(d) f ‘(2) = – 1, f (3) = 1, f (x) → 3 as x → ∞
Applied Calculus ,4/E, Deborah Hughes-Hallett
Copyright 2010 by John Wiley and Sons, All Rights Reserved