Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics, and probability / Continuous mathematics / 4 / a

Table of contents
- i a
  - Solution i
- ii a
  - Solution ii

i

Solution

The second-derivative criterion for convexity is

f is convex on R ⟺ f^{''} (x) \geq 0 for every x \in R .

The non-strict inequality is necessary and sufficient; the stronger condition

f^{''} > 0

is sufficient for strict convexity.

ii

Solution

Expand before taking the second derivative:

f (x) = e^{3 x} + 3 e^{x} + 3 e^{- x} + e^{- 3 x}, f^{''} (x) = 9 e^{3 x} + 3 e^{x} + 3 e^{- x} + 9 e^{- 3 x} > 0.

The second-derivative criterion for convexity proves that $f$ is strictly convex, hence convex.

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