Math 1210
Quiz 6B
February 21st, 2014
Answer the following two (2) questions. The value of every question is
indicated at the beginning of it. Please show your work (derivatives, study of
signs, etc. Since the instructions are long you may use an additional sheet of
paper.
Name:
1. (15 points) Consider the function f (x) =
UID:
3x5 −20x3
32
over (−∞, +∞).
1. (2 points) What is the domain of f ?
2. (4 points) Where is f increasing (resp. decreasing)? Namely, for which x is f 0 (x)
positive (resp. negative)?
3. (4 points) What are, if any, the inflection points?
4. (3 points) Where is f concave up (resp. down)? Namely, for which x is f 00 (x)
positive (resp. negative)?
5. (2 points) Are there any local extreme points?
2. (15 points) Find the equation of the line that is tangent to the circle x2 + y 2 = 1 in
the first quadrant and forms with the coordinate axes the triangle with smallest possible
area. You can follow the following steps.
1. Fix a point (a,b) lying on the circle x2 + y 2 = 1 and find the equation of the tangent
line to the circle at (a, b).
Hint: In the first quadrant, the circle is given by
√
y = f (x) = + 1 − x2
You can either compute the slope by differentiating this function or use implicit
differentiation.
2. Determine the points of intersection of this line with the coordinate axes. This will
give you the triangle side lengths, and hence its area.
3. At this point you should have an expression for the area√as a function of a and b,
but since (a, b) lies on the circle, you actually have b = 1 − a2 , so you can write
the area as function of a only. Write it down, and convince yourself that the domain
of this area function A(a) is (0, 1).
4. Find the local extreme points of the area function A(a) over (0, 1). Namely, compute
A0 (a), determine for which values of a you have A0 (a) = 0 (you should find only
one) and finally conclude that the point you found is indeed a minimum.
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