Chapter 4 | Applications of Derivatives
377
104. Absolute maximum at x =2 and absolute minima at x =±3 105. Absolute minimum at x =1 and absolute maximum at x =2 106. Absolute maximum at x =4, absolute minimum at x =−1, local maximum at x =−2, and a critical point that is not a maximum or minimum at x =2 107. Absolute maxima at x =2 and x =−3, local minimum at x =1, and absolute minimum at x =4 For the following exercises, find the critical points in the domains of the following functions.
123. y = x +sin( x ) over [0, 2 π ] 124. y = x 1+ x over [0, 100] 125. y = | x +1 | + | x −1| over [−3, 2] 126. y = x − x 3 over [0, 4] 127. y = sin x +cos x over [0, 2 π ] 128. y =4sin θ −3cos θ over [0, 2 π ]
For the following exercises, find the local and absolute minima and maxima for the functions over (−∞, ∞). 129. y = x 2 +4 x +5 130. y = x 3 −12 x 131. y =3 x 4 +8 x 3 −18 x 2 132. y = x 3 (1− x ) 6 133. y = x 2 + x +6 x −1 134. y = x 2 −1 x −1 For the following functions, use a calculator to graph the function and to estimate the absolute and local maxima and minima. Then, solve for them explicitly. 135. [T] y =3 x 1− x 2 136. [T] y = x +sin( x ) 137. [T] y =12 x 5 +45 x 4 +20 x 3 −90 x 2 −120 x +3 138. [T] y = x 3 +6 x 2 − x −30 x −2 139. [T] y = 4− x 2 4+ x 2 140. A company that produces cell phones has a cost function of C = x 2 −1200 x +36,400, where C is cost in dollars and x is number of cell phones produced (in thousands). How many units of cell phone (in thousands) minimizes this cost function?
108. y =4 x 3 −3 x 109. y =4 x − x 2 110. y = 1 x −1 111. y = ln( x −2) 112. y = tan( x ) 113. y = 4− x 2 114. y = x 3/2 −3 x 5/2
x 2 −1 x 2 +2 x −3
115. y =
116. y = sin 2 ( x )
117. y = x + 1 x
For the following exercises, find the local and/or absolute maxima for the functions over the specified domain. 118. f ( x ) = x 2 +3 over [−1, 4] 119. y = x 2 + 2 x over [1, 4]
2
⎛ ⎝ x − x 2
⎞ ⎠
120. y =
over [−1, 1]
121. y = 1 ⎛
over (0, 1)
⎞ ⎠
⎝ x − x 2
122. y = 9− x over [1, 9]
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