# Exponential and Logarithmic Functions: $y=2e^{-5x^2}$

Use the function $$y=2e^{-5x^2}$$

I have already answered most of the questions, but I would really appreciate it if you would look over my answers and tell me if I am wrong and help me correct my mistakes. I mostly need help with e) and g)

a) State the domain:

I wrote domain as $x\in\mathbb R$.

b) Determine the intercepts, if any:

No x-intercept, and y-intercept at $(0,2)$.

c) Discuss the symmetry of the graph:

I said the graph is symmetric with respect to the y-axis because the function is even.

d) Find any asymptotes:

No vertical asymptote, horizontal asymptote at $y=0$.

e) Determine the intervals of increase and decrease:

I know I'm supposed to set the first derivative to zero then solve, but I get lost and I need help. How do I find the intervals of increase and decrease?

f) What is the maxima and/or minima:

Maxima is $2$ at $x=0$, and minima value does not exist.

g) Where is the curve concave upward or downward:

I know to set the second derivative to zero, but again, I get lost, I need help on this one.

h) Locate the points of inflection: Not sure if it's correct, but I got

$$\left(-\frac{1}{\sqrt{10}},\frac{2}{\sqrt e}\right) \text{ and } \left(\frac{1}{\sqrt {10}}, \frac{2}{\sqrt e}\right)$$

• What graph are you looking at? Just insert $x=0$ into $y=e^{-5x^2}$ and you will see y is not $2$ when $x=0$. – randomgirl Jan 20 '16 at 4:50
As for the increase/decrease thing, $f'$ will tell you this information. You already found when $f'=0$. Use test numbers on both sides of $x=0$ to see if $f'>0$ (this means $f$ is increasing) or if $f'<0$ (this means $f$ is decreasing). You can do something similar with the concavity thing. You found when $f''=0$ Just choose test numbers for each of the three intervals to see if $f''>0$ (this means $f$ is concave up) or if $f''<0$ (this means $f$ is concave down). If concavity switches, at the points where $f''=0$ then at these points you have an inflection point.