{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

4.4 Review Questions

4.4 Review Questions - Solutions to Review Problems for the...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Solutions to Review Problems for the Final Exam Solutions to Maxima, minima, convexity, concavity 1. f ( x ) = 2 3 x 3- x 2- 4 x + 1, f ( x ) = 2 x 2- 2 x- 4, f ( x ) = 4 x- 2 (a) f ( x ) = 0 when 0 = x 2- x- 2 = ( x + 1)( x- 2), i.e., x =- 1, x = 2. (b) f (- 1) =- 6 < 0, maximum; f (2) = 6 > 0 minimum. (c) increasing for x <- 1, x > 2, (d) convex for x > 1 / 2. 2. f ( x ) = 2 x + 18 /x , f ( x ) = 2- 18 /x 2 , f ( x ) = 36 /x 3 (a) f ( x ) = 0 when 2 = 18 /x 2 , i.e., x = 3, x =- 3 (b) f (- 3) =- 36 / 27 < 0, maximum; f (3) = 36 / 27 > 0, minimum (c) increasing for x <- 3, x > 3; (d) convex for x > 3. f ( x ) = x x 2 +1 , f ( x ) = ( x 2 +1)- x · 2 x ( x 2 +1) 2 = 1- x 2 ( x 2 +1) 2 (a) f ( x ) = 0 when x 2 = 1, i.e., x =- 1, x = 1; (c) increasing for- 1 < x < 1; (b) f ( x ) =- 2 x ( x 2 +1) 2- (1- x 2 ) · 2( x 2 +1)(2 x ) ( x 2 +1) 4 f (- 1) = 2 / 2 2 > 0, minimum; f (1) =- 2 / 2 2 < 0, maximum (d) f ( x ) =- 2 x ( x 2 +1)(3- x 2 ) ( x 2 +1) 4 , convex for- √ 3 < x < 0, x > √ 3 4. f ( x ) =- 2 x- 1 / 2- 3 ln x + 4 √ x , f ( x ) = x- 3 / 2- 3 /x + 2 x- 1 / 2 , (a) f ( x ) = x- 1 / 2 ( x- 1- 3 x- 1 / 2 + 2) = 0 when x- 1 / 2 = 1 , 2 or x = 1 , x = 1 / 4. (b) f ( x ) =- (3 / 2) x- 5 / 2 + 3 x- 2- x- 3 / 2 , f (1) =- (3 / 2) + 3- 1 > 0 minimum f (1 / 4) =- (3 / 2)(32) + 3 · 16- 8 =- 8 < 0 maximum (c) increasing for x < 1 / 4 and x > 1; (d) convex when x- 3 x 1 / 2 + (3 / 2) < 0, that is when r 1 < √ x < r 2 where r i = (3 ± √ 9- 6) / 2 5. f ( x ) = x ( x- 1) 4 , f ( x ) = ( x- 1) 4 + 4 x ( x- 1) 3 = (5 x- 1)( x- 1) 3 f ( x ) = 0 when x = 1 / 5, x = 1. f is increasing for x < 1 / 5, decreasing on (1 / 5...
View Full Document

{[ snackBarMessage ]}

Page1 / 6

4.4 Review Questions - Solutions to Review Problems for the...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online