1. AP Calculus
4.2 Extreme Values

2. Applications of the Derivative
• One of the most common applications of the
derivative is to find maximum and/or minimum
values of a function
• These are called “Extreme Values” or “Extrema”
• Extrema would be an excellent name for an 80’s
Hair Band.

3. Definition
• Let f (x) be a function defined on an interval
I, let a ϵ I, then f (a) is
• Absolute minimum of f (x) on I, if
f (a) ≤ f (x) for all x in I
• Absolute maximum of f (x) on I, if
f (a) > f (x) for all x in I
• If no interval is indicated, then the extreme
values apply to the entire function over its
domain.

4. Do All Functions Have Extrema?
• f (x) = x
• No extrema unless the function is defined on an
interval.

5. Do All Functions Have Extrema?
• g (x) = (-x(x2 – 4))/x
• Discontinuous and has no max on [a, b]
a b

6. Do All Functions Have Extrema?
• f (x) = tan x
• No max or min on the open interval (a, b)
a b

7. Do All Functions Have Extrema?
• h(x) = 3x3 + 6x2 + x + 3
• Function is continuous and [a, b] is closed.
Function h(x) has a min and max.
b
a

8. Definition
Local Extrema a function f (x) has a:
• Local Minimum at x = c if f (c) is the
minimum value of f on some open interval (in
the domain of f) containing c.
• Local Maximum at x = c if f (c) is the
maximum value of f on some open interval
containing c.

9. Local Max and Min
Local Max
Local Min
Local Min

10. Absolute and Local Max
(a, f(a)) (c, f (c))
Absolute max on [a,b] (b, f(b))
Local Max
a c b

11. Critical Points
Definition of Critical Points
• A number c in the domain of f is called a critical
point if either f’ (c) = 0 or f’ (c) is undefined.

12. Fermat’s Theorem
• Theorem: If f (c) is a local min or
max, then c is a critical point of f.
• Not all critical points yield local extrema. “False
positives” can occur meaning that f’(c) = 0 but
f(c) is not a local extremum.

13. Fermat’s Theorem
f(x) = x3 + 4
Tangent line at (0, 4) is horizontal
f(0) is NOT an extremum

14. Optimizing on a Closed Interval
Theorem: Extreme Values on a Closed
Interval
• Assume f (x) is continuous on [a, b] and let f(c)
be the minimum or maximum value on [a, b].
Then c is either a critical point or one of the
endpoints a or b.

15. Example
Find the extrema of f(x) = 2x3 – 15x2 + 24x + 7 on
[0, 6].
• Step 1: Set f’(x) = 0 to find critical points
▫ f’(x) = 6x2 – 30x + 24 = 0, x = 1, 4
• Step 2: Calculate f(x) at critical points and
endpoints.
▫ f(1) = 18, f(4) = -9, f(0) = 7, f(6) = 43
• The maximum of f(x) on [0, 6] is (6, 43) and
minimum is (4, -9).

16. Graph of f(x)=2x3-15x2+24x+7
Endpoint
Max (6, 43)
Critical Point – local
max (1, 18)
Endpoint
(0, 7)
Critical point – local min
(4, -9)

17. Example
• Compute critical points of
• Find extreme values on [0, 2]

18. Example
• Critical point: h’(t) = 0, t = 0
• Local min (0, -1)
• Endpoints: (-2, 1.44), (2, 1.44) maximums

19. Example
• Find the extreme values of g(x) = sin x cos x on
[0, π]

20. Example
• Critical points: g’(x) = cos 2 x – sin 2 x
• g’(x) = 0, x = π/4, 3π/4
• g(π/4) = ½ , max
• g(3π/4) = -1/2 , min
• Endpoints (0, 0), (π, 0)

21. Rolle’s Theorem
• Assume f (x) is continuous on [a, b] and
differentiable on (a, b). If f (a) = f (b) then there
exists a number c between a and b such that
f’(c) = 0
f(c)
f(a) f(b)
a c b

22. Example
• Use Rolle’s Theorem to show that the function
f(x) = x3 + 9x – 4 has at most 1 real root.

23. Example
• If f (x) had 2 real roots a and b, then f (a) = f (b)
and Rolle’s Theorem would apply with a number
c between a and b such that f’(c) = 0.
• However…f’(x) = 3x2 + 9 and 3x2 + 9 = 0 has no
real solutions, so there cannot be a value c such
that f’ (c) = 0 so there is not more than 1 real
root of f (x).