The figure below shows the graph of $f(x) = x^5-8x^3+1$
The two marked points are at a local max and min.
The curve in the figure has one point whose tangent line is horizontal even though the point itself is neither a max nor a min. Find the (x,y) coordinates of that point.
If the tangent line is horizontal, doesn't that mean that the derivative is 0 at that point?
So i need to find at which point the derivative is 0?
$f'(x)= 5x^4-24x^2$
$0=x^2(5x^2-24)$
$x=0$ is the point since the other two points are the max and min. Plugging this back into $f(x)$ gives 1, so the point is $(0,1)$.
Is this correct?
Best Answer
Yes. This is basic calculus. Here are the graphs:
An inflection point occurs when the first and second derivatives vanish and that is indeed at $(0,1)$.
[Click on animation]