Problem 1.

(a) Find two functions f and g satisfying f(0) = 0, g(0) = 0 and x = f(x)g(x).

(b) Prove that there do not exist two differentiable functions f and g with f(0) = 0, g(0) = 0 and x = f(x)g(x).

Exam 1 : Due: Tuesday, March 11 2

Problem 2. Suppose that f : [0, 1] → [0, 1] is differentiable and that f′(x) ≠ 1 for any x. Prove that there is one and only one number c ∈ [0, 1] with f(c) = c.

Problem 3. Suppose a < b < c < d < e. Where is the absolute minimum of (a) f(x) = (x − a)^{2} + (x − b)^{2} + (x − c)^{2} + (x − d)^{2} + (x − e)^{2}

(b) g(x) = |x − a| + |x − b| + |x − c| + |x − d| + |x − e|

(c) Prove that if a + \frac{b}{2} + \frac{c}{3} + \frac{d}{4} + \frac{e}{5} = 0 then a + bx + cx^{2} + dx^{3} + ex^{4} = 0 for some x ∈ [0, 1].

Exam 1 : Due: Tuesday, March 11 4

Problem 4. Here’s a picture of two lines tangent to the elliptic curve y^{2} = x^{3} −5x+2. Where do these lines intersect?

EXAM

Exam 1

Math 157

Due: Tuesday, March 11

• This exam is due at the beginning of class on Tuesday, October 8.

• You are allowed to use your book or your notes, but you may not consult with any person about the exam, or use the internet as a resource for the exam.

• Each part of each problem is worth one point for a total of twelve possible points.

Success!