CHAPTER 4 APPLICATIONS OF DERIVATIVES

Transcription

1 CHAPTER 4 APPLICATIONS OF DERIVATIVES 4. EXTREME VALUES OF FUNCTIONS. An asolute minimum at c, an asolute maimum at. Theorem guarantees the eistence of such etreme values ecause h is continuous on [aß ]. 2. An asolute minimum at, an asolute maimum at c. Theorem guarantees the eistence of such etreme values ecause f is continuous on [aß ].. No asolute minimum. An asolute maimum at c. Since the function's domain is an open interval, the function does not satisfy the hypotheses of Theorem and need not have asolute etreme values. 4. No asolute etrema. The function is neither continuous nor defined on a closed interval, so it need not fulfill the conclusions of Theorem. 5. An asolute minimum at a and an asolute maimum at c. Note that y g() is not continuous ut still has etrema. When the hypothesis of Theorem is satisfied then etrema are guaranteed, ut hen the hypothesis is not satisfied, asolute etrema may or may not occur. 6. Asolute minimum at c and an asolute maimum at a. Note that y g() is not continuous ut still has asolute etrema. When the hypothesis of Theorem is satisfied then etrema are guaranteed, ut hen the hypothesis is not satisfied, asolute etrema may or may not occur. 7. Local minimum at a ß!, local maimum at aß! 8. Minima at a ß! and aß!, maimum at a!ß 9. Maimum at a!ß &. Note that there is no minimum since the endpoint aß! is ecluded from the graph. 0. Local maimum at a $ß!, local minimum at aß!, maimum at aß, minimum at a!ß. Graph (c), since this the only graph that has positive slope at c. 2. Graph (), since this is the only graph that represents a differentiale function at a and and has negative slope at c.. Graph (d), since this is the only graph representing a funtion that is differentiale at ut not at a. 4. Graph (a), since this is the only graph that represents a function that is not differentiale at a or.

2 20 Chapter 4 Applications of Derivatives f() 5 Ê f () Ê no critical points; f( 2), f() Ê the asolute maimum is at and the asolute minimum is 9 at 2 6. f() 4 Ê f () Ê no critical points; f( 4) 0, f() 5 Ê the asolute maimum is 0 at 4 and the asolute minimum is 5 at 7. f() Ê f () 2 Ê a critical point at 0; f( ) 0, f(0), f(2) Ê the asolute maimum is at 2 and the asolute minimum is at 0 8. f() % Ê f () 2 Ê a critical point at 0; f( ) 5, f(0) 4, f() Ê the asolute maimum is 4 at 0 and the asolute minimum is 5 at $ 9. F() Ê F () 2, hoever 2 $ 0 is not a critical point since 0 is not in the domain; F(0.5) 4, F(2) 0.25 Ê the asolute maimum is 0.25 at 2 and the asolute minimum is 4 at 0.5

3 Section 4. Etreme Values of Functions F() Ê F (), hoever 0 is not a critical point since 0 is not in the domain; F( 2), F( ) Ê the asolute maimum is at and the asolute minimum is at 2 $ 2. h() È Î$ Î$ Ê h () Ê a critical point at 0; h( ), h(0) 0, h(8) 2 Ê the asolute maimum is 2 at 8 and the asolute minimum is at Î$ Î$ 22. h() Ê h () Ê a critical point at 0; h( ), h(0) 0, h() Ê the asolute maimum is 0 at 0 and the asolute minimum is at and at 2. g() È4 a4 Î Ê g () 4 Î a ( 2) È 4 Ê critical points at 2 and 0, ut not at 2 ecause 2 is not in the domain; g( 2) 0, g(0) 2, g() È Ê the asolute maimum is 2 at 0 and the asolute minimum is 0 at g() È5 Î Î a& a5 ( 2) Ê g () ˆ Ê critical points at È5 È & and 0, ut not at È5 ecause È5 is not in the domain; f Š È5 0, f(0) È5 Ê the asolute maimum is 0 at È5 and the asolute minimum is È5 at 0 ˆ ˆ ˆ 5 6 ) at ) 25. f( )) sin ) Ê f ()) cos ) Ê ) is a critical point, ut ) is not a critical point ecause is not interior to the domain; f, f, f Ê the asolute maimum is at minimum is and the asolute

4 22 Chapter 4 Applications of Derivatives 26. f( )) tan ) Ê f ()) sec ) Ê f has no critical points in ˆ ß 4. The etreme values therefore occur at the endpoints: f ˆ È and f ˆ Ê the asolute 4 maimum is at ) 4 and the asolute minimum is È at ) 27. g() csc Ê g () (csc )(cot ) Ê a critical point at ; g ˆ 2, g ˆ, g ˆ 2 2 Ê the È È 2 2 È asolute maimum is at and, and the asolute minimum is at 28. g() sec Ê g () (sec )(tan ) Ê a critical point at 0; g ˆ 2, g(0), g ˆ 2 Ê the asolute 6 maimum is 2 at and the asolute minimum is at 0 È È Î 29. f(t) 2 kk t t at Ê f (t) t t t a (2t) Î Èt Ê a critical point at t 0; f( ), f(0) 2, f() Ê the asolute maimum is 2 at t 0 and the asolute minimum is at t kk t È Î 0. f(t) kt 5 k (t 5) a(t 5) Ê f (t) a (t 5) (2(t 5)) t 5 t 5 Î È(t 5) kt 5k Ê a critical point at t 5; f(4), f(5) 0, f(7) 2 Ê the asolute maimum is 2 at t 7 and the asolute minimum is 0 at t 5

5 Section 4. Etreme Values of Functions 2. ga e Ê g a e e Ê a critical point at ; ga e, and ga e, Ê the asolute maimum is e at and the asolute minimum is e at 2. The first derivative h a has no zeros, so e need only consider the endpoints. ha0 ln ; ha ln 4 Maimum value is ln 4 at ; Minimum value is ln at The first derivative f a has a zero at. Critical point value: fa ln Endpoint values: fa0.5 2 ln ; fa4 ln 4.66; Asolute maimum value is 4 ln 4 at 4; Asolute Minimum value is at ; Local maimum at ˆ 2 ln 2 ß ga e Ê g a 2e Ê a critical point at 4 0; ga 2 e, ga0, and ga e Ê the asolute maimum is at 0 and the asolute 4 minimum is e at 2 %Î$ 4 Î$ 5. f() Ê f () Ê a critical point at 0; f( ), f(0) 0, f(8) 6 Ê the asolute maimum is 6 at 8 and the asolute minimum is 0 at 0 &Î$ 5 Î$ 6. f() Ê f () Ê a critical point at 0; f( ), f(0) 0, f(8) 2 Ê the asolute maimum is 2 at 8 and the asolute minimum is at $Î& Î& 7. g( )) ) Ê g ()) 5 ) Ê a critical point at ) 0; g( 2) 8, g(0) 0, g() Ê the asolute maimum is at ) and the asolute minimum is 8 at ) 2 Î$ Î$ 8. h( )) ) Ê h ()) 2 ) Ê a critical point at ) 0; h( 27) 27, h(0) 0, h(8) 2 Ê the asolute maimum is 27 at ) 27 and the asolute minimum is 0 at ) 0

6 24 Chapter 4 Applications of Derivatives 9. Minimum value is at. 40. To find the eact values, note that y $, hich is zero hen % È' $ *. Local maimum at É $ Š É ß % a 0Þ86ß 5Þ089 ; local % È' $ * minimum at Š É ß% a0.86ß To find the eact values, note that that y $ ) % a$ % a, hich is zero hen 2 or $. Local maimum at a ß 7 ; local minimum at ˆ % ß % $ ( 42. Note that y $ ' $$ a, hich is zero at. The graph shos that the function assumes loer values to the left and higher values to the right of this point, so the function has no local or gloal etreme values. 4. Minimum value is 0 hen or.

7 Section 4. Etreme Values of Functions The minimum value is at!. 45. The actual graph of the function has asymptotes at, so there are no etrema near these values. (This is an eample of grapher failure.) There is a local minimum at a!ß. 46. Maimum value is 2 at ; minimum value is 0 at and $. 47. Maimum value is at à minimum value is as. 48. Maimum value is at 0à minimum value is as 2.