Question

How does extremas and points of inflection change, if the function changes from `y=f(x)` to `y=Af(x)`, where `A>0` is a constant?

Answer 1

Please see below.

Explanation:

Assume `y=Af(x)`, where `A` is a constant.

Then `(dy)/(dx)=Af'(x)` and we have a maxima at `Af'(x)=0`

and dividing by `A` at `f'(x)=0`.

What about point of inflection? This is given by `(d^2y)/(dx^2)=0`. For `y=f(x)`, it is at `f''(x)=0`.

As `y=Af(x)`, we have `(d^2y)/(dx^2)=Af''(x)=axx0=0`

Hence it is apparent that as `A` changes the only thing which happens is that the function get stretched / shrunk along `y`-axis, with no change in the nature of curve. For example see the graph of `y=xe^(-2x)` and `y=3xe^(-2x)`.

graph{xe^(-2x) [-0.401, 2.099, -0.585, 0.665]}

graph{3xe^(-2x) [-0.401, 2.099, -0.585, 0.665]}