Following Question 2, can we determine whether the function is convex, concave, or neither?

Question

BlackTom AI · Accepted Answer

To assess convexity or concavity of a function in this context, we should examine what each option implies and whether it aligns with standard criteria for convexity/concavity.
Option 1: 'We must evaluate the function at many values of x ∈ R^2 × 1 to draw a conclusion.' This phrasing is unclear and seems to reference evaluating on a specific domain or a miswritten space (R^2 × 1). In general, convexity/concavity can be studied via second derivatives, epigraphs, or line tests, not necessarily by sampling a large number of points alone. Therefore, as stated, this option does not provide a valid or complete method for determining convexity or concavity.
Option 2: 'The function is concave.' This asserts a definitive property of the function. To justify concavity, one would need to show that the function lies below its tangents or that the Hessian is negative semidefinite (in the differentiable case) or apply a one-variable/ multivariable concavity test. Without performing such a check, stating concavity is a claim that requires evidence.
Option 3: 'The function is convex.' Similar to Option 2, asserting convexity requires verification, such as demonstrating that the function lies above its tangents or that the Hessian is positive semidefinite. Merely naming convexity without supporting analysis is insufficient.
Option 4: 'The function is neither convex nor concave.' This option claims that the function does not satisfy either standard property globally. To support this, one would typically construct a counterexample showing a line segment on which the function lies above (or below) the function values in a way that violates the defining inequality for convexity or concavity, or show Hessian indefiniteness or region-dependent behavior. Without such demonstration, this remains a possibility but not a proven conclusion.
In summary, each option relies on a different path to justification: Option 1 critiques the method, Options 2 and 3 require formal verification of concavity/convexity, and Option 4 requires a counterexample or region-wise analysis. The problem statement references 'Following Question 2' and asks whether we can determine the classification; the appropriate approach would be to apply a standard test (second derivative or Hessian, or line test) to reach a definitive conclusion. The provided answer option asserts concavity in Option 2, which would be supported only if the concavity criteria are met. The reasoning for why Option 2 is correct would hinge on demonstrating negative semidefinite Hessian (or equivalent concavity test) across the domain. The reasoning for why Options 1, 3, and 4 are incorrect hinges on either lack of proper methodology, or on asserting a property without verification, or on presenting a global classification that would require counterevidence.}

25S-STATS-102B-LEC-3 S25 Midterm Exam- Requires Respondus LockDown Browser

Following Question 2, can we determine whether the function is convex, concave, or neither?

View Explanation

Log in for full answers

Similar Questions

Suppose 𝑔 ( 𝑤 | 𝑦 𝑖 , 𝑥 𝑖 ) = ( 𝑦 𝑖 − 𝑥 𝑖 𝑇 𝑤 ) 2 , where 𝑤 ∈ 𝑅 𝑝 , 𝑦 𝑖 ∈ 𝑅 , and 𝑥 𝑖 ∈ 𝑅 𝑝 , for 𝑖 = 1 … 𝑛 . Is ∑ 𝑖 = 1 𝑛 𝑔 ( 𝑤 | 𝑦 𝑖 , 𝑥 𝑖 ) convex?

𝑔 ( 𝑤 ) = − 𝑐 𝑜 𝑠 ( 2 𝜋 𝑤 𝑇 𝑤 ) + 𝑤 𝑇 𝑤 , where 𝑤 ∈ 𝑅 𝑁 . Is 𝑔 ( 𝑤 ) convex?

Given C below, what is the correspoding shape of g(w)?

Given C below, what is the correspoding shape of g(w)?

What is the nature of this cost function? Check all that apply.

Which of these plots shows a Non-Convex function? (select all that apply)

In a consumer society, many adults channel creativity into buying things

Economic stress and unpredictable times have resulted in a booming industry for self-help products

More Practical Tools for International Students

Homework AI Solver

Stylized Writing Assistant

Plagiarism Checker Assistant

Citation Assistant

In-Class Translation Assistant

Class Notes Assistant

Quiz Search Assistant

Past Exam Questions by School

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler