Question

It is given that the roots of the equation $x^2-4x-{\log }_{3}P=0$ are real. For this, the minimum value of $P$ is

• A. $\frac{1}{27}$
• B. $\frac{1}{64}$
• C. $\frac{1}{81}$
• D. $1$

Answer 1

Answer: (C)

$\frac{1}{81}$

Given quadratic equation is $x^2-4x-{\log }_{3}P=0$,

As the roots of the quadratic equation are real the value of the discriminant of the quadratic equation $\Delta \ge 0$,

$⟹16+4{\log }_{3}P\ge 0⟹{\log }_{3}P\ge -4⟹P\ge \frac{1}{81}$,

$\therefore$ the minimum value of $P$ is $\frac{1}{81}$