A student studies the cubic f(x) = x³ + ax² + 3x + 7, where a is a real number…

A student studies the cubic $f(x) = x^3 + ax^2 + 3x + 7$, where $a$ is a real number. She wants $f$ to be strictly increasing on the whole real line, with no interval where the graph falls.

Find $f'(x)$. Write the condition on $a$ that keeps $f'(x)$ from ever being negative. Then give the largest and the smallest value of $a$ that work. At each of these two extreme values, $f'(x)$ becomes the square of a bracket times a number. Say why $f$ is still strictly increasing there, even though $f'$ is $0$ at one point.

Check your answer in two ways. Take $a = 0$, which lies in your range, and show that $f'(x)$ is positive for every $x$. Take $a = 4$, which lies outside it, and find a value of $x$ at which $f'(x)$ is negative.