Let f(x)=e^(3x)-kx, for k>0.
Using a calculator or computer sketch the graph of f for k=1/9,1/6,1/3,1/2,1,2,4. Describe what happens as k changes.
1.f(x) has a local minimum. Find the location of the minimum.
2. Find the y-coordinate of the minimum.
3. Find the value of k for which this y-coordinate is the largest.
4. How do you know that this value of k maximizes the y-coordinate? Find d^2y/dk^2 to use the second-derivative test.
f' = 3e^(3x) - k
so it has a minimum when x = 1/3 ln(k/3)
to find k where the y-coordinate is largest, note that this coordinate is
e^(3(1/3 ln(k/3)) - k(1/3 ln(k/3))
= e^(ln(k/3)) - k/3 ln(k/3)
= k/3 (1 - ln(k/3))
which is a maximum at k=3
yo can check the 2nd derivative
For the second derivative, I got 9e^(3x) but it says its wrong. Am I missing a step?
Find a formula for a curve of the form y=e−(x−a)2/b
for b>0 with a local maximum at x=6 and points of inflection at x=3 and x=9.
To graph the function f(x) = e^(3x) - kx for different values of k, you can use a calculator or a computer with graphing software. Here's how you can do it step by step:
1. Open a graphing calculator or graphing software on your computer.
2. Set up the coordinate system with the x-axis and y-axis.
3. Plot the points for different values of x, ranging from negative infinity to positive infinity.
4. For each value of x, calculate the corresponding y-value using the function f(x) = e^(3x) - kx, where k takes different values.
5. Plot these points on the graph.
6. Connect the plotted points to form a smooth curve.
Now, let's answer the questions based on the graph:
1. To find the location of the local minimum, look for the lowest point on the graph. This point represents the x-coordinate of the local minimum.
2. Once you've identified the x-coordinate of the local minimum, substitute it into the function f(x) = e^(3x) - kx to find the y-coordinate.
3. To find the value of k for which the y-coordinate is the largest, observe the graph and identify the value of k that leads to the highest point on the curve.
4. To determine whether this value of k maximizes the y-coordinate, you can use the second-derivative test by finding d^2y/dk^2. If the second derivative is negative, then it is a maximum point. Evaluate d^2y/dk^2 and determine if it is negative at the value of k obtained in question 3.
Note that these calculations might require algebraic manipulation and differentiation.