Critical Points. Local Maxima and Minima

Calculus and Vectors – How to get an A+

4.2 Critical Points. Local Maxima and Minima

A Local Maximum B Local Minimum A function f has a local (relative) maximum at x = c if A function f has a local (relative) minimum at x = c if f (x) ≤ f (c) when x is sufficiently close to c (from both f (x) ≥ f (c) when x is sufficiently close to c (from both sides). sides). f (c) is called the local (relative) maximum value. f (c) is called the local (relative) minimum value. (c, f (c)) is called the local (relative) maximum point. (c, f (c)) is called the local (relative) minimum point.

C Note Ex 1. A function is defined by the following graph. Find the local minimum or maximum points. The following points are neither local minimum or maximum points.

Local minimum points are A(−5,2) , B(−1,1) , and C(5,2) . Local maximum points are D(−2,4) and E(2,5) .

D Critical Numbers and Critical Points Ex 2. Find the critical points for: 3 2 The number c ∈ D f is a critical number if either a) f (x) = 2x + 3x f '(c) = 0 or f '(c) DNE f '(x) = 6x2 + 6x = 6x(x +1) The point P(c, f (c)) is called critical point. f '(x) = 0 ⇒ x = 0 or x = −1 f (0) = 0 f (−1) = 2(−1)3 + 3(−1)2 = −2 + 3 =1 The critical points are A(0,0) and B(−1,1) .

b) f (x) = 3 x2 −1 ' 1 2x f '(x) = [(x2 −1)1/3 ] = (x2 −1)−2/3 (2x) = 3 3 3 (x2 −1)2 f '(x) = 0 ⇒ x = 0, y = −1 f '(x) DNE ⇒ x2 −1 = 0 ⇒ x = ±1, y = 0 The critical points are A(0,−1) , B(−1,0) , and C(1,0) . 4.2 Critical Points. Local Maxima and Minima ©2010 Iulia & Teodoru Gugoiu - Page 1 of 2 Calculus and Vectors – How to get an A+

E Fermat’s Theorem Note. If f has a local extremum (minimum or maximum) at If x = c then c is a critical number for f . So, at a local f '(c) = 0 or f '(c) DNE extremum: then the function f may have or not a local extremum f '(c) = 0 or f '(c) DNE (minimum or maximum) at x = c .

F First Derivative Test Ex 3. Find the local extrema points for the functions in Let c be a critical number of a continuous function f . the previous example. If f '(x) changes sign from positive to negative at 3 2 c then f has a local maximum at c . a) f (x) = 2x + 3x

f '(x) = 6x2 + 6x = 6x(x +1) Sign Chart:

x −1 0 f (x) _ 1 ` 0 _ f '(x) + 0 - 0 +

If f '(x) changes sign from negative to positive at The point A(0,0) is a local minimum point and the point c then f has a local minimum at c . B(−1,1) is a local maximum point.

b) f (x) = 3 x2 −1

2x f '(x) = 3 3 (x2 −1)2

Sign Chart: If f '(x) does not change sign at c then f has no local extremum (minimum or maximum) at c . x −1 0 1 f (x) ` 0 ` −1 _ 0 _ f '(x) - DNE - 0 + DNE +

The point A(0,−1) is a local minimum point.

Ex 4. Find a function of the form f (x) = ax4 + bx2 + cx + d with a local maximum at (0,−6) and a local minimum at

(1,−8) .

f (0) = −6 ⇒ d = −6 (1) f (1) = −8 ⇒ a + b + c + d = −8 (2) f '(x) = 4ax3 + 2bx + c f '(0) = 0 ⇒ c = 0 (3) f '(1) = 0 ⇒ 4a + 2b + c = 0 (4) ⎧a + b + 0 − 6 = −8 ⎧a + b = −2 (1),(2),(3),(4) ⇒ ⎨ ⇒ ⎨ ⇒ ⎩4a + 2b + 0 = 0 ⎩2a + b = 0 a = 2 and b = −4 ∴ f (x) = 2x4 − 4x2 − 6