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Math 1A: Derivative Tests and Optimization 1. Find the intervals of increasing and decreasing local max's and min's, intervals of concavity, and the inflection points (if any) of the given function a) y --2-3x+r b) f(x)=x+3x 2. The sum of two positive numbers is 16. What is the smallest possible value of the sum of their squares? 3. Find the dimensions of a rectangle with area 1000 m^2 whose perimeter is as small as possible. 4. A box with an open top is to be constructed from a square piece of cardboard, 6 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. 5. An open topped serving box will be made by cutting squares out of each corner of a 12" by 18" sheet of cardboard and folding the tabs up to form a box. What size squares should be cut out to maximize the volume of the box 6. Find the dimensions of the biggest rectangle that can be inscribed inside of a 6 cm, 8 cm, 10 cm right triangle. 7. Find the critical numbers of the following function: ()-1'+P+r+1 8. Use calculus to find the dimensions of a rectangle with perimeter 100 m whose area is as large as possible. 9. Find the point on the line y = 2x + 3 that is closest to the origin. 10. Find the point on the curve y -Vx that is closest to the point (3.0)