Find all points on the graph of f(x)=xe^x at which the tangent line is parallel to the line y-3x=4.
i figured out that f'(x)=3 but i don't know what to do with that.
To find the points on the graph of f(x) = xe^x where the tangent line is parallel to the line y - 3x = 4, you need to determine the value of x that satisfies two conditions: the derivative of f(x) should be equal to the slope of the given line, and the corresponding y-coordinate should satisfy the equation of the line.
Let's break down the process step-by-step:
1. Find the derivative of f(x):
The derivative of f(x) is f'(x). You mentioned that you have already found f'(x) = 3, which means that the slope of the tangent line to f(x) at any point is 3.
2. Determine the slope of the given line:
The given line is y - 3x = 4. Rewrite it in slope-intercept form (y = mx + b) to identify the slope. Adding 3x to both sides and rearranging, we get y = 3x + 4. The slope of this line is 3, which matches the slope of the tangent line.
3. Set the derivative equal to the slope of the line:
Since the slope of the tangent line is 3, set f'(x) = 3 and solve for x.
3 = 3
x^2e^x + xe^x = 3 (since f'(x) = x^2e^x + xe^x)
x^2e^x + xe^x - 3 = 0
This is a non-linear equation, and solving it directly can be challenging. You might need to use numerical or graphical methods to find x.
4. Once you find the value(s) of x, substitute it back into the equation for f(x) to find the corresponding y-coordinate(s):
For each value of x you obtain from step 3, substitute it back into f(x) = xe^x to find the corresponding y-coordinate(s).
These (x, y) coordinates will give you the points on the graph of f(x) = xe^x where the tangent line is parallel to the line y - 3x = 4.