Mathematics 1 - Mock Midterm 2
Exercise 1. Consider
f (x, y) = x3 − 3xy 2
a) Find one restriction passing through (0, 0) where f is neither convex nor
concave.
b) Find the stationary (or critical) points.
c) Is it possible to determine the nature of the critical points through the Hessian Matrix?
d) If not, can you infer the nature of the point by a)?
Exercise 2. Given the function
f (x, y) = 3x3 + 2y 3 + 3xy 2 − 3x + 1.
a) Determine the critical points and study their nature.
b) Does the function have a global maximum and a global minimum?
c) Find the linear approximation of f at (0, −1). Is f differentiable at (0, −1)?
If yes, explain the geometric interpretation of differentiability.
d) Given the region
D = {(x, y) ∈ R2 | y ≤ 1, y ≤ x ≤ 2y}
find the absolute maximum and minimum of f on D (if any).
Exercise 3. Consider the function
Z x2 +3
F (x) =
e
dt
.
ln t
a) Find the Domain of F (x) and explain how to compute derivative of F (x).
1
b) Find the linear and quadratic approximation about the point x0 = e.
Exercise 4. Compute the following integral
Z 2
1
ln4 (x)
dx.
x
2