Practice problem

Finding Concavity and Points of Inflection Graphically

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Which of the following inequalities are true for all positive $x_1$ and $x_2$ because of the algebraic definition of a concave function?

Choose all correct answers

$\ln\left(\frac{1}{2}x_1 + \frac{1}{2}x_2\right) \geq \frac{1}{2}\ln(x_1) + \frac{1}{2}\ln(x_2)$

$\ln\left(\frac{1}{3}x_1 + \frac{1}{3}x_2\right) \geq \frac{1}{2}\ln(x_1) + \frac{1}{2}\ln(x_2)$

$\ln\left(\frac{1}{3}x_1 + \frac{2}{3}x_2\right) \geq \frac{1}{3}\ln(x_1) + \frac{2}{3}\ln(x_2)$

$\ln\left(\frac{1}{3}x_1 + \frac{2}{3}x_2\right) \geq \ln(x_1) + \ln(x_2)$