ISI2024-MCS-PCA | Question: 16

Question

Let $n \geq 2$. Suppose $\alpha_1, \alpha_2, \cdots, \alpha_n$ are real numbers such that $\alpha_1<\alpha_2<\cdots<\alpha_n$. Let
$$
P(x)=\left(x-\alpha_1\right)^2\left(x-\alpha_2\right) \cdots\left(x-\alpha_n\right), x \in \mathbb{R} .
$$

If $P^{\prime}$ is the derivative of $P$, then $P^{\prime}(x)$ has

• A. no real root.
• B. $n$ real roots, the smallest of which is $\alpha_1$.
• C. $n-1$ real roots, the smallest of which is $\alpha_1$.
• D. $n$ real roots, the smallest of which is strictly larger than $\alpha_1$.