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GO Classes | IIITH PGEE 2026 Mock Test 12 | Question: 48

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Consider the family of functions
$$
g(x)=a \ln (x)+\frac{b}{x}
$$
defined for $x>0,$ where $a$ and $b$ are positive constants.

Any function $g(x)$ in this family has only one critical point. In terms of $a$ and $b,$ what is the $x$-coordinate of that critical point?

  1. $x=b / a$ is a critical point where the function shows local maxima
  2. $x=a / b$ is a critical point where the function shows local minima
  3. $x=b / a$ is a critical point where the function shows local minima
  4. $x=a / b$ is a critical point where the function shows local maxima

1 Answer

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Detailed Video Solution: https://youtu.be/6aIuj2b2J38 

Since
$$
g^{\prime}(x)=\frac{a}{x}-\frac{b}{x^2}=\frac{a x-b}{x^2}
$$
we see that $g^{\prime}(x)=0$ when $a x-b=0$, i.e. when $x=\dfrac{b}{a}$.
We use the $2^{\text{nd}}$ derivative test. Since
$$
g^{\prime \prime}(x)=-\frac{a}{x^2}+\frac{2 b}{x^3}=\frac{-a x+2 b}{x^3}
$$
we have that
$$
g^{\prime \prime}\left(\frac{b}{a}\right)=\frac{-a \frac{b}{a}+2 b}{\left(\frac{b}{a}\right)^3}=\frac{(-b+2 b) a^3}{b^3}=\frac{a^3}{b^2}>0 . \quad {\color{Red}{\text {(local minima)}}}
$$

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