A function from dice

We have the derivative of the function as $3x^2+2ax+b$ . The function itself is increasing for all x if and only if its derivative is nonnegative for all x. Since the derivative's leading coefficient is positive, this is equivalent to requiring that it's determinant is nonpositive, i.e. less than or equal to 0. This boils down to the condition $a^2 \leq 3b$ . Note that the condition does not depend at all on the outcome of the roll of the third die, so we can take the sample space below to be the 36 distinct outcomes from rolling the first two dice.

To figure the probability $P(a^2 \leq 3b)$ , we can easily enumerate the 36 possibilities. For example, if $a=1$ , it follows that $a^2=1 \leq 3b$ for all $b \in \{1,2,3,4,5,6\}$ . Working similarly with other outcomes for a, it can be easily shown that there are 16 combinations of a and b that satisfy the requirement. So, $P(a^2 \leq 3b) = \frac{16}{36} = \frac{4}{9}$ , making the required answer 13.