Question

Let $p\left(x\right)=x^2+ax+b$ have two distinct real roots, where $a,b$ are real numbers. Define $g\left(x\right)=p\left(x^3\right)$ for all real numbers $x$. Then which the following statements are true?
I. $g$ has exactly two distinct real roots
II. $g$ can have more than two distinct real roots
III. There exists a real number $\alpha$ such that $g\left(x\right)\ge \alpha$ for all real $x$

• A. Only I
• B. Only I and III
• C. Only II
• D. Only II and III

Answer 1

The correct option is B Only I and III

Given that $p\left(x\right)=x^2+ax+b.$
Let $p\left(x\right)$ has two roots as $\alpha ,\beta$

$p\left(x^3\right)=0\Rightarrow x^3=\alpha \text{or}{x}^3=\beta$
$x^3=\alpha \text{and}{x}^3=\beta$ has only one real solution each.
So, $g\left(x\right)=p\left(x^3\right)=0$ has exactly two real solutions.

$\underset{x\to -\infty }{lim}g\left(x\right)=\underset{x\to \infty }{lim}g\left(x\right)\to \infty$
As $g\left(x\right)$ is a continuous function, it will have global minima.
Therefore, Statement III is correct.