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Magazine Problem 55
Sketch a continuous curve that has the given characteristics. $$f(1)=0 ; f^{\prime}(x)>0 \text { for all } x ; f^{\prime \prime}(x)<0 \text { for all } x$$
Problem 56
A cylindrical cup (no top) is designed to hold \(375 \mathrm{cm}^{3}(375 \mathrm{mL})\) There is no waste in the material used for the sides. However, there is waste in that the bottom is made from a square \(2 r\) on a side. What are the most economical dimensions for a cup made under these conditions?
Problem 56
In Exercises \(55-57\), sketch a continuous curve that has the given characteristics. \(f(0)=1 ; f^{\prime}(x) < 0\) for all \(x ; f^{\prime \prime}(x) < 0\) for \(x < 0 ; f^{\prime \prime}(x) > 0\) for \(x > 0\)
Problem 56
Sketch a continuous curve that has the given characteristics. \(f(0)=1 ; f^{\prime}(x)<0\) for all \(x ; f^{\prime \prime}(x)<0\) for \(x<0 ; f^{\prime \prime}(x)>0\) for \(x>0\)
Problem 56
Solve the given maximum and minimum problems. A cylindrical cup (no top) is designed to hold \(375 \mathrm{cm}^{3}(375 \mathrm{mL}).\) There is no waste in the material used for the sides. However, there is waste in that the bottom is made from a square \(2 r\) on a side. What are the most economical dimensions for a cup made under these conditions?
Problem 57
In Exercises \(55-57\), sketch a continuous curve that has the given characteristics. \(f(-1)=0 ; f(2)=2 ; f^{\prime}(x) < 0\) for \(x < -1 ; f^{\prime}(x) > 0\) for \(x > -1 ; f^{\prime \prime}(x) < 0\) for \(0 < x < 2 ; f^{\prime \prime}(x) > 0\) for \(x < 0\) or \(x > 2\)
Problem 57
Sketch a continuous curve that has the given characteristics. $$\begin{aligned} &f(-1)=0 ; f(2)=2 ; f^{\prime}(x) < 0 \text { for } x < -1 ; f^{\prime}(x) > 0 \text { for }\\\ &x > -1 ; f^{\prime \prime}(x) < 0 \text { for } 0 < x < 2 ; f^{\prime \prime}(x) > 0 \text { for } x<0 \text { or }\\\ &x > 2 \end{aligned}$$
