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domain and range

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Suppose that $f(x)$ and $g(x)$ are functions on $\mathbb{R}$ such that the range of $f$ is $[-5,3]$, and the range of $g$ is $[-2,1]$. The range of $f(x) \cdot g(x)$ is $[a,b]$. What is the largest possible value of $b$?

Feb 25, 2021

We want to maximize $xy$ if $x\in[-5,3]$ and $y\in [-2,1]$. This either occurs when they are both negative or both positive. If both positive, then it's $3*1=3$. If both negative, it's $(-5)*(-2)=10$. Since $10>3$, the answer is $\boxed{10}$.