Assignment #2
Notes: Same rule applies. If you work in a group, please write the name/ID of the members and
how much they contributed towards the completion. Please work in a separate sheet of paper
with clearly eligible handwriting. If you have any questions, please let me know in advance.
Deadline: 04/07/2023, 11:59 a.m. (A minute before class time)
Q1: Partial Derivatives & Differentials
Find 𝑓 , 𝑓 , 𝑓 , 𝑓 , 𝑓 , 𝑎𝑛𝑑 𝑓
in each of the following.
1. 𝑓(𝑥, 𝑦) = 𝑥 𝑦 + 5𝑦 − 𝑥 + 7
2. 𝑓(𝑥, 𝑦) = cos(𝑥𝑦 ) + sin(𝑥)
3. 𝑓(𝑥, 𝑦) = 𝑒
√𝑥 + 1
Make sure to clearly indicate which differentiation rule or property was used in each step.
Q2: Properties of Continuous Functions
Please provide a proof to each of the below.
Suppose that 𝑓, 𝑔: 𝑋 → 𝑅 are continuous functions, then


(𝑓 ± 𝑔) is continuous
(𝑓 ⋅ 𝑔) is continuous
Suppose that 𝑔: 𝑋 → 𝑅 and 𝑓: 𝑔[𝑋] → 𝑅 are continuous functions, then

(𝑓 ∘ 𝑔) is continuous
Q3: Linear Combination of Concave Function
Suppose 𝑓: 𝑋 → 𝑅 is a concave function. Using just the definition of concavity, prove that if 𝑎 ∈
𝑅 and 𝛽 > 0, then 𝑔: 𝑋 → 𝑅 defined by 𝑔(𝑥) = 𝑎 + 𝛽𝑓(𝑥) is also a concave function.
Bonus Questions
Q1: Inverse Function Theorem, part 1
Is the following statement true? Provide a proof to support your argument.
If 𝑋 ⊂ 𝑅 and 𝑓: 𝑋 → 𝑅 is strictly monotonic, then 𝑓
: 𝑓[𝑋] → 𝑌 exists.
Q2: Inverse Function Theorem, part 2
Is the following statement true? Provide a proof to support your argument.
If 𝑓 exists, is 𝑓: 𝑋 → 𝑅 necessarily monotonic? If not, what additional assumption on 𝑓 is
required for the statement to be true.
Q3: Intermediate Value Theorem – Corollary
Please provide a formal proof for the following corollary.
Suppose X is an interval and 𝑓: 𝑋 → 𝑅 is continuous. Then 𝑓[𝑋] is an interval.
Hint:
1) Start by considering any 𝑦, 𝑤 ∈ 𝑓[𝑋], where 𝑦 ≠ 𝑤.
2) Choose 𝑎, 𝑏 ∈ 𝑋 with 𝑎 < 𝑏 such that 𝑓(𝑎) = 𝑦 𝑎𝑛𝑑 𝑓(𝑏) = 𝑤.
3) Then, apply the IVT.
Use concise math/English to summarize how you arrived at this conclusion.