One-one

Calculus Level 2

We know that the function $f(x)=e^x$ is injective on $R$ and that its derivative is always positive at all points.

Is it true that any function whose derivative vanishes at least at one point is many-to-one on the interval that contains those points?

Always true Sometimes true Never true

1 solution

Janardhanan Sivaramakrishnan
Feb 19, 2018

Consider the two functions $f_1(x)=x^2$ and $f_2(x)=x^3$ in the interval $(-1,1)$

The derivatives of both functions vanish at $x=0$ . However, while $f_1(x)$ is many to one in the said interval ( $f(-1/2)=f(1/2)=1/4$ ), $f_2(x)$ is a one to one function in the interval, eventhough the derivative vanishes at a point in the interior of the interval.

One-one

1 solution

0 pending reports