
Verify that the tangent line is horizontal.

Answer to Problem 68E
Tangents drawn on the function f(x)=xe−x are horizontal.
Explanation of Solution
Given information:
Use the function and its derivative to determine any points on the graph of f at which the tangent line is horizontal. Use a graphing utility to verify the results.
f(x)=xe−x , f'(x)=e−x−xe−x
Calculation:
Consider the function, f(x)=xe−x
The slope of the function or the derivative is f'(x)=e−x−xe−x
The slope of the tangent drawn on the function f(x)=xe−x should be horizontal.
So, the slope will be 0
−xe−x−e−x=0e−x(−x+1)=0
x=
0 , 1
Now, x=
0 , 1 should satisfy the function.
When, x=
0
f(0)=0×e−0
=0
When, x=
1
f(1)=e−1
So, the points at which the function f(x)=xe−x and tangents drawn it is horiontal are (0,0) and (1,e−1) .
Now to check the above point by calculator, proceed as follows.
Press button "=Y" and then enter the expression in the window as given below.
Now, press the "WINDOW" button to set the range.
Now, press the "GRAPH" button to plot the graph as for different point .
For point (0,0)
For point (1,e−1)
Hence, it is verified that the tangents drawn on the function f(x)=xe−x are horizontal.
Chapter 12 Solutions
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