Can exponential graphs be concave up decreasing?

Answer 1

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.