Lesson 66 — Concavity and Inflection Points

Determine the concavity of $f(x) = x^2$ on all of $\mathbb{R}$. Is there an inflection point?

Determine the concavity and inflection points of $f(x) = x^3$.

Concavity of $f(x) = x^4$. Is there an inflection point at $x = 0$? Justify using the sign of $f''$.

Concavity of $f(x) = e^x$ on all of $\mathbb{R}$. Is there an inflection point?

Concavity of $f(x) = \ln x$ on $(0, \infty)$.

Concavity of $f(x) = \sin x$ on $[0, 2\pi]$. Identify the inflection points.

Concavity of $f(x) = \cos x$ on $[0, 2\pi]$. Inflection points.

Concavity of $f(x) = 1/x$ on the intervals $(0,\infty)$ and $(-\infty,0)$.

Concavity of $f(x) = e^{-x^2/2}$ (Gaussian). Identify the inflection points.

Concavity and inflection of $f(x) = x^3 - 3x^2 + 2$.

Use the $f''$ test: classify the extrema of $f(x) = x^3 - 12x$.

Extrema of $f(x) = x^4 - 4x^2$ via $f''$ test.

Extrema of $f(x) = x e^{-x}$ via $f''$.

Extrema of $f(x) = x^2 \ln x$ on $(0, \infty)$ via $f''$.

Extrema of $f(x) = \sin x + \frac{1}{2}\sin(2x)$ on $[0, 2\pi]$.

Show that $f(x) = x^4$ has a minimum at $x = 0$ despite $f''(0) = 0$ (inconclusive test).

Show that $f(x) = x^5$ has no extremum at $x = 0$ despite $f'(0) = 0$.

For $f(x) = x^2 + 1/x$ on $x > 0$: find the minimum and justify with $f''$.

Extrema of $f(x) = \ln x / x$ on $(0, \infty)$ via $f''$.

Extrema of $f(x) = x^{1/x}$ on $(0, \infty)$ (take $\ln f$ before differentiating).

Cost $C(q) = q^3 - 6q^2 + 9q + 100$. Find the inflection point and interpret it as a change in marginal return.

Profit $\pi(q) = -q^3 + 30q^2 - 100q$. Maximize via $\pi'$ and confirm with $\pi''$.

Logistic curve $P(t) = K/(1 + e^{-rt})$. Show there is an inflection point at $P = K/2$ (half the carrying capacity).

Potential energy $U(x) = -\cos x$ (pendulum). Find stable and unstable equilibria using $U''$.

Harmonic spring: $U(x) = \frac{1}{2}kx^2$. Show $x = 0$ is a stable equilibrium using $U''$.

Bernoulli entropy $H(p) = -p\ln p - (1-p)\ln(1-p)$. Show $H'' < 0$ and that the maximum is at $p = 1/2$.

Learning curve $L(t) = 1 - e^{-kt}$. Determine its concavity. What does it say about the speed of learning?

In an epidemic, the peak of new cases occurs at the inflection point of the cumulative case curve $f(t)$. Justify geometrically and via $f''$.

Utility $U(W) = \ln W$ is concave. Explain how Jensen's inequality implies risk aversion for this investor.

Why does the linear regression loss function have a unique global minimum? Justify using convexity.

What is the correct condition for $x_0$ to be an inflection point of $f$?

Prove that the sum of two convex functions is convex, using the definition via $f''$.

Show that $f$ convex on $I$ implies the midpoint inequality: $f\!\left(\frac{x+y}{2}\right) \leq \frac{f(x)+f(y)}{2}$.

Why is $f''(x_0) = 0$ not sufficient to guarantee an inflection point? Give a concrete counterexample.

Show that $\ln$ is concave on $(0,\infty)$ and use it to prove the AM-GM inequality: $(x+y)/2 \geq \sqrt{xy}$ for $x, y > 0$.

Huber function $L(x) = x^2/2$ if $|x| \leq 1$; $|x| - 1/2$ otherwise. Is it convex? Where is $L''$ discontinuous?

Prove the second derivative test using the Taylor polynomial of order 2.

Prove Jensen's inequality for two points: $f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2)$ — directly from the definition of convexity.

Prove that a convex function on an open interval is continuous on the interior.

Prove that $f$ is convex if and only if its graph lies above every tangent line: $f(y) \geq f(x) + f'(x)(y-x)$ for all $x, y$.