4. Find any x and y intercepts, asymptotes, horizontal, vertical, or otherwise, find where the function assumes positive values and where it assumes negative values, the places where the first derivative is either 0 or does not exists, where the function is increasing and decreasing, find the places where the second derivative is 0 or doesn't exist, and where the function is concave up and concave down. Find the local maxima and minima and points of inflection, and sketch the graph of
y  intercept: let x = 0
x  intercept.
Exponential functions always take on positive values and are never 0, so there is not x  intercept.
Since e to more negative values becomes smaller and smaller, the x  axis is a horizontal asymptote.
We are ready to consider the first derivative to look for max's and mins.
To differentiate this function, we use the chain rule.Apply the derivative of the outside function to the inside function and multiply that by the derivative of the inside function. The outside function is the exponential function, and it is equal to its derivative, so when we apply the derivative of the outside function to the inside function, we just get the original function. However, the chain rule tells us that we need to multiply this by the derivative of the inside function. The inside function is the exponent in the exponential function, and this is a polynomial. Multiply the power times the coefficient and raise the variable to one smaller power. The coefficient is 1/(2s^{2}).
The 2's cancel out and this cleans up to
y' = 0 when
Set the factors = 0


so the only place where y' = 0 is when x = 0.
Since the second factor is always positive, y' > 0 when x 0, so f(x) is increasing when x < 0, and y' < 0 when x > 0, so f(x) is decreasing when x > 0.
Thus f(x) has a max when x = 0 because it stops increasing and starts decreasing.
We next consider f"(x) to look for points of inflection. Since y' is given by a product, we use the product rule to find y"
When we set y" = 0, we factor
and set the factors = 0


If x < s, then y" > 0 and the function is concave up.
If s < x < s, then y" < 0 and the function is concave down.
If x > s, the y" > 0and the function is concave up.
As a result, the function has a points of inflection when x = s and when x = s.
This gives us the following points to plot
The point is that we do not need to know what s is. Whatever it is, the points of inflection will be found at x = s and x = s, and the y values will always be e^{1/2}.
The graph looks like
This is essentially the graph of the bell shaped curve. The graph of the bell shaped curve has a constant out in front in order to make the total area under the graph be equal to 1 for reasons which you will see if you take a course in probability theory, and it has an x minus the mean where we have just an x. s is the standard deviation. An interesting thing about the bell shaped curve is that it has points of inflection one standard deviation above and below the mean.