Consider the polynomial $p(x) = (x + a_1)(x + a_2) \cdots (x + a_{10})$ where $a_i$ is a real number for each $i = 1, \ldots, 10$. Suppose all of the eleven coefficients of $p(x)$ are positive. For each of the following statements, decide if it is true or false. Write your answers as a sequence of four letters (T/F) in correct order. (i) The polynomial $p(x)$ must have a global minimum. (ii) Each $a_i$ must be positive. (iii) All real roots of $p'(x)$ must be negative. (iv) All roots of $p'(x)$ must be real.
Consider the polynomial $p(x) = (x + a_1)(x + a_2) \cdots (x + a_{10})$ where $a_i$ is a real number for each $i = 1, \ldots, 10$. Suppose all of the eleven coefficients of $p(x)$ are positive. For each of the following statements, decide if it is true or false. Write your answers as a sequence of four letters (T/F) in correct order.\\
(i) The polynomial $p(x)$ must have a global minimum.\\
(ii) Each $a_i$ must be positive.\\
(iii) All real roots of $p'(x)$ must be negative.\\
(iv) All roots of $p'(x)$ must be real.