# How to find the approximate position of any roots of an original function when given the graph of a derivative function?

If you were given the graph of a derivative function, how would you go about finding the approximate roots of the original function?

I've been able to find the stationary points, points of inflection etc. using the graph of this derivative function in order to draw a sketch of the original function but I'm unable to figure out a way to get an approximation of the roots using the derivative function graph. Once you draw the original function, and seeing where it cuts the x-axis, would you just do it by eye?

## 2 Answers

The derivative $f'(x)$ doesn't tell you anything about the location of the roots of $f(x)$. In general, $f'(x)$ only tells you about how $f(x)$ is changing, not about what $f(x)$ is at a given point.

For example, say $f(x) = x^2$, $g(x) = x^2 + 4$, $h(x) = x^2 - 4$. $f(x)$ has one root at $x=0$ while $g(x)$ has no roots and $h(x)$ has two roots. However, they all have the same derivative: $f'(x) = g'(x) = h'(x) = 2x$. So, just knowing the derivative there is no way to determine the roots of the original function.

You can't because you don't know the constant of integration. Say your derivative is $2x$. Then the function is $x^2+c$. If $c$ is positive, there are no real roots at all. If $c$ is large and negative, the roots are large and one of each sign. You can get the shape of the original function, but you don't know where to locate it vertically on the paper.