Question 1014593: Please help me solve this problem, I have 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 the mistakes. I mostly need help with e) and g).
Use the function 2e^(-5x^2) to answer each question.
a) State the domain:
I wrote the domain as x is an element of all real numbers
b) Determine the intercepts, if any:
I found that there was no x-intercept, and a 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:
I found no vertical asymptote, and a horizontal asymptote at y=0.
e) Determine the intervals of increase and decrease:
I need help with this.
f) What is the maxima and/or minima:
Maxima is 2 at x=0, no minima value.
g) Where is the curve concave upward and downward:
I need help with this.
h) Locate the points of inflection:
I don't know if this is right or wrong but i calculated them as
( -(1/sqrt(10)), 2/sqrt(e) ) and ( 1/sqrt(10), 2/sqrt(e) )
i) Graph the function:
I've already graphed it.
Thank-you, your help is always appreciated!
Found 2 solutions by stanbon, solver91311:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Please help me solve this problem, I have 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 the mistakes. I mostly need help with e) and g).
Use the function 2e^(-5x^2) to answer each question.
--------------------------------
a) State the domain:
I wrote the domain as x is an element of all real numbers:: OK
----------------------------
b) Determine the intercepts, if any:
I found that there was no x-intercept, and a y-intercept at (0,2):: OK
----------------------------
c) Discuss the symmetry of the graph:
I said the graph is symmetric with respect to the y-axis because the function is even.:: OK
---------------------------------------
d) Find any asymptotes:
I found no vertical asymptote, and a horizontal asymptote at y=0.:: OK
---------------------------------------
e) Determine the intervals of increase and decrease:: Look at the graph,
or determine the 1st derivative, set to zero and solve.
I need help with this.
f) What is the maxima and/or minima::OK
Maxima is 2 at x=0, no minima value.:OK
g) Where is the curve concave upward and downward:
Find the 2nd derivative to determine the answer or
just look at the graph.
h) Locate the points of inflection:: Use 2nd derivative
I don't know if this is right or wrong but i calculated them as
( -(1/sqrt(10)), 2/sqrt(e) ) and ( 1/sqrt(10), 2/sqrt(e) )
i) Graph the function:: Done
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Cheers,
Stan H.
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Answer by solver91311(24713)  (Show Source):
You can put this solution on YOUR website!
Everything you did answer is spot-on correct.
Part e
Intervals of increase/decrease. Intuitively, an interval of increase is any interval where the function gets bigger as you go from left to right. An interval of decrease is an interval where the function gets smaller as you move from left to right. Precisely speaking, A function is increasing on an interval if, and only if, for all pairs of numbers  in the interval, \ <\ f(x_2)) . The way to find the intervals is to find the intervals where the first derivative is positive and the intervals where the first derivative is negative.
So we need the solution interval for:
But since  for the entire domain, we only have to solve:
Which, as a quick examination of the graph would suggest, is the interval ) .
Similarly, your function decreases on )
In point of fact, since the terms "increasing" and "decreasing" are only defined on an interval, you can't actually exclude a single point in the definition of that interval. See discussion: Open or Closed? by Lin McMullin
Based on that, your intervals should be: Increasing on  ] and Decreasing on [)
Part g
Concavity is determined by the sign of the second derivative.
So you need to solve:
for the two concave up intervals, and
for the concave down interval.
As you might suspect, the critical points are the two inflection points that you found in Part h.
Here is the graph to compare with your effort:
John
My calculator said it, I believe it, that settles it
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