[14 points] Let $\mathbb{R}_+$ denote the set of positive real numbers. For a continuous function $f: \mathbb{R}_+ \rightarrow \mathbb{R}_+$, define $A_r =$ the area bounded by the graph of $f$, X-axis, $x = 1$ and $x = r$ $B_r =$ the area bounded by the graph of $f$, X-axis, $x = r$ and $x = r^2$. Find all continuous $f: \mathbb{R}_+ \rightarrow \mathbb{R}_+$ for which $A_r = B_r$ for every positive number $r$. Hints (use these or your own method): Find an equation relating $f(x)$ and $f(x^2)$. Consider the function $xf(x)$. Suppose a sequence $x_n$ converges to $b$ where all $x_n$ and $b$ are in the domain of a continuous function $g$. Then $g(x_n)$ must converge to $g(b)$. E.g., $g\left(3^{\frac{1}{n}}\right) \rightarrow g(1)$.
[14 points] Let $\mathbb{R}_+$ denote the set of positive real numbers. For a continuous function $f: \mathbb{R}_+ \rightarrow \mathbb{R}_+$, define\\
$A_r =$ the area bounded by the graph of $f$, X-axis, $x = 1$ and $x = r$\\
$B_r =$ the area bounded by the graph of $f$, X-axis, $x = r$ and $x = r^2$.\\
Find all continuous $f: \mathbb{R}_+ \rightarrow \mathbb{R}_+$ for which $A_r = B_r$ for every positive number $r$.\\
Hints (use these or your own method): Find an equation relating $f(x)$ and $f(x^2)$. Consider the function $xf(x)$. Suppose a sequence $x_n$ converges to $b$ where all $x_n$ and $b$ are in the domain of a continuous function $g$. Then $g(x_n)$ must converge to $g(b)$. E.g., $g\left(3^{\frac{1}{n}}\right) \rightarrow g(1)$.