Find the smallest number b such that the function

Question

Find the smallest number b such that the function $f(x)=x^3+7x^2+bx+4$

is invertible. Evaluate $\frac{\mathrm{d}}{\mathrm{d}x}(f^{-1})(4)$ using that $b$.

Answer 1

HINT: To make it invertible, you have to make it one-to-one. This cubic is increasing when $x$ is very negative or very positive, so the way to do this is to make it increasing. Thus, you want the smallest $b$ that ensures that $f\,'(x)\ge 0$ for all $x$. Its graph will then look like that of $y=x^3$, rising to a single point with a horizontal tangent and then rising again, instead of rising, falling, and then rising again.