Can exponential graphs be concave up decreasing?
Yes. A simple example is the graph of the function y=e-x This is
a simple manipulation of a basic exponential function, y=ex.
Graphically it is obvious that this function is decreasing and
concave upward across it entire domain, but it is easy to show this
mathematically as well
For, the function y=e -x, the basic pattern for exponential
derivatives, as well as the chain rule, will supply the first and
second derivatives of this function, which will be necessary to
mathematically determine the concavity and "slope" of this
function. I will assume you know how to do basic derivatives since
you are asking this question. You seem to be well entrenched in
typical curriculum for a derivative calculus course. I will list
the original function and its first and second derivatives
below:
f(x)=e-x
f'(x)=-e-x
f''(x)=e-x
When the first derivative is examined, you will notice that
regardless of what number is input for the variable "x", the result
will always be negative, which proves that the function will remain
decreasing for its entirety.
When the second derivative is examined, you will notice that
regardless of what number is input for the variable "x", the result
will always be positive. This proves that the function will remain
upwardly concave for its entirety.
You can also use the fact that if f(x)=ex then f(-x) = e-x and
we have created a reflection across the y axis since that is what
f(-x) does. The reflection does not change the fact the graph is
concave up, but it does show it is decreasing. The fact that the
second derivative is greater than 0 confirms this. As you remember,
is says if f"(x) is greater than 0 for all x on some interval I,
then f(x) is concave up on I.