Critical Points, Local Maxima, and Local Minima
Find all critical points, local minima, and
local maxima of the following functions.
f ( x) =
4 x 3 + 3 x 2 − 36 x − 5
What are the critical points of ?
What are the local minima of ?
What are the local maxima of ?
How do the above answers change if the
upper hole at 3 is filled in?
How do the above answers change if the
lower hole at 3 is filled in?
The table records the rate of change of air
temperature, H, as a function of hours since
midnight, t, during one morning.
t
dH / dt
6 7 8 9 10 11 12
1 2 0 -2 0 3 2
When was the temperature a local minimum?
Local maximum?
g ( x) =
x − 2 ln( x 2 + 3)
Critical Points, Local Maxima, and Local Minima
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.
(b) Temperature 50◦ at 10:00 am and 40◦ at
2:00 pm.
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 → ∞
(c) Temperature rising at 10:00 am and falling
at 2:00 pm.
(d) Temperature 50◦ at 10:00 am and 2:00
pm, 60◦ at noon.
(e) Temperature 50◦ at 10:00 am and 60◦ at
2:00 pm.
If the graph is that of f '( x) , state true or false
for the following statements about 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 max. and one is a local min.
(c) The derivative is zero at two values of x,
one is a local max. while the other is neither a
local max. nor a min
(d) The derivative is zero at two values of x,
one is a local min. while the other is neither a
local max. nor a min.
(e) The derivative is zero only at one
value of x where it is a local min.
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
→∞