Function machine-1

Algebra Level 2

$\large f(x)=\frac{x+a}{\sqrt{x^2+b}}$ $a$ and $b$ both are positive.

For which value of $x$ is $f(x)$ the largest?

\frac{a^2}{b}

\frac{\sqrt{a}}{b}

\frac{a}{\sqrt{b}}

\sqrt{\frac{a}{b}}

\frac{b}{a}

\frac{a^2}{b^2}

\frac{a}{b}

\frac{a}{b^2}

1 solution

A Former Brilliant Member
Nov 3, 2019

Let $y=f(x)=\dfrac{x+a}{\sqrt {x^2+b}}$ . Then $x^2(y^2-1)-2ax+by^2-a^2=0$ . Since $x$ is real, therefore discriminant of this equation must be non-negative definite. This yields $y_{max}=\sqrt {\dfrac{a^2+b}{b}}$ when $x=\dfrac{a}{y^2-1}=\dfrac{b}{a}$

Function machine-1

1 solution

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