Final answer:
The points on the graph of the function f(x) = x/(x+2) where the tangent has a slope of 1/2 are (0,0) and (-4,2), obtained by finding the derivative of the function, setting it equal to 1/2, and solving for x.
Step-by-step explanation:
To determine which points (x,y) on the graph of the function f(x) = x/(x+2) have a tangent line with slope 1/2, we need to find the derivative of the function, set it equal to 1/2, and solve for x. The derivative of the function f(x) gives us the slope of the tangent line at any point on the graph.
Derivative of f(x):
f'(x) = d/dx [x/(x+2)] which simplifies to f'(x) = 2/(x+2)^2.
We then set the derivative equal to 1/2 to find the x-values where the slope of the tangent is 1/2: 2/(x+2)^2 = 1/2.
Solving for x gives us two solutions: x = 0 and x = -4.
Plugging these back into the original function, we get the corresponding y-values; f(0) = 0/2 = 0 and f(-4) = -4/(-4+2) = -4/-2 = 2.
Therefore, the points on the graph where the tangent has a slope of 1/2 are (0,0) and (-4,2), which corresponds to option c.
Complete Question:
the function f is defined by f(x) = x/x+2 what points (x,y) on the graph of f have the property that the line tangent to f at (x,y) has slope 1/2?
a. (0,0) only
b. (1/2,1/5) only
c. (0,0) and (-4,2)
d.(0,0) and (4,2/3)
e. there are no such points