Question

Is `f(x)=e^x` increasing or decreasing at `x=1`?

Answer 1

Increasing.

Explanation:

We can use the derivative of a function to determine if a function is increasing or decreasing at a point:

• If `f'>0` at `x=a`, then `f` is increasing at `x=a`.
• If `f'<0` at `x=a`, then `f` is decreasing at `x=a`.

We have:

`f(x)=e^x`

And the derivative of `e^x` is itself:

`f'(x)=e^x`

We see that:

`f'(1)=e^1=e`

Since `f'(1)>0`, we see that `f` is increasing at `x=1`.

We can check a graph of `f(x)`:

graph{e^x [-8.92, 11.08, -2.48, 7.52]}

In fact, since `f'(x)=e^x`, we can see that this derivative will always be positive, and that the function `e^x` is increasing for all Real values of `x`.