Mathematics - Probability Question with Solution | TestHub

The function $f(x)$ is increasing, which means $f^{\prime }(x)\ge 0$ for all $x$. Also, the condition $f^{\prime }(x)=0$ should not form an interval.

We have $f^{\prime }(x)=3x^2+2ax+b$.

Therefore, $a^2-3b\le 0$.

This is true for exactly 16 ordered pairs $(a,b)$ where $1\le a,b\le 6$:

(1,1), (1,2), (1,3), (1,4), (1,5), (1,6)

(2,2), (2,3), (2,4), (2,5), (2,6)

(3,3), (3,4), (3,5), (3,6)

(4,6)

Thus, the probability $P(E)=\frac{16}{36}=\frac{4}{9}$.