Capital One·Data Scientist·Technical Phone Screen·Intermediate

Intermediate

Jun 2026

Summary

Capital One data scientist interview with a pretty heavy math-based question that felt more like a calculus exam than anything I expected from a DS role. One question, lots of moving parts.

Questions Asked (1)

Given a twice-differentiable function with specific limit behavior at infinity, known sign patterns for its first and second derivatives, and a point where the second derivative is undefined, sketch a qualitatively correct graph marking all local extrema and inflection points, determine whether two local maxima and one local minimum are consistent with the given conditions, and identify where inflection points must occur along with the slope behavior around the discontinuity in the second derivative.

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

I was not expecting pure calculus in a data science screen.

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AI HintsAI Generated

Suggested Approach

First, translate the given limit behavior and derivative sign patterns into a qualitative shape: increasing/decreasing intervals and concavity. Then, systematically identify local extrema (where f' changes sign) and inflection points (where f'' changes sign or is undefined), and check consistency with the proposed extrema count. Finally, sketch the graph, marking all critical points and ensuring the slope behavior around the discontinuity in f'' aligns with the sign of f'.

Pro tip: Always justify each feature of the graph by referencing the sign of f' and f''; this shows rigorous reasoning and avoids guesswork. Also, explicitly state that a local extremum requires f' to change sign, not just be zero.

1. Interpret limit behavior and derivative signs

Use the limits at infinity to determine horizontal asymptotes or end behavior. From the sign of f', determine where the function is increasing or decreasing; from the sign of f'', determine concavity (concave up/down).

2. Identify local extrema and inflection points

Local maxima occur where f' changes from positive to negative; local minima where f' changes from negative to positive. Inflection points occur where f'' changes sign or is undefined (with concavity change).

3. Check consistency of proposed extrema

Verify whether two local maxima and one local minimum can coexist given the sign patterns of f'. For example, if f' is positive then negative then positive then negative, that yields two maxima and one minimum. Ensure the overall behavior matches the limits.

4. Sketch the graph with key points

Draw a smooth curve that respects the increasing/decreasing intervals, concavity, and the point where f'' is undefined (which may be a cusp or vertical tangent). Mark all local extrema and inflection points clearly.

5. Analyze slope behavior at f'' discontinuity

At the point where f'' is undefined, the slope (f') may have a local extremum or change behavior. Determine whether f' is continuous and whether its sign changes, which affects whether that point is an extremum or just an inflection point.

Key Points to Mention

The relationship between the sign of f' and increasing/decreasing behavior, and between f'' and concavity.
The first derivative test for local extrema: sign change of f' is necessary.
The second derivative test and its limitations when f'' is zero or undefined.
Inflection points require a change in concavity, which may occur where f'' is zero or undefined.
The importance of checking consistency with limits at infinity (e.g., horizontal asymptotes).
How a discontinuity in f'' can create a sharp turn or cusp, affecting the slope behavior.

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.