For which point(s) on the graph of f(x)=5x/(x+2) is the slope of the tangent 2/5?
a) (-3,15)
b) (7, 35/9)
c) (3,3) and (-7,7)
d) (24, 60/13)
Answer is C?
You are correct
To find the point(s) on the graph of f(x)=5x/(x+2) where the slope of the tangent is 2/5, we need to find the derivative of the function and set it equal to 2/5. Then, we can solve for x to find the corresponding y-coordinate.
Let's start by finding the derivative of f(x). The quotient rule can be used to differentiate f(x) = 5x/(x+2):
f'(x) = [(d/dx)(5x)(x+2) - (5x)(d/dx)(x+2)] / (x+2)^2
= (5(x+2) - 5x) / (x+2)^2
= 5(2) / (x+2)^2
= 10 / (x+2)^2
Now, we set f'(x) = 2/5 and solve for x:
10 / (x+2)^2 = 2/5
To simplify the equation, we can cross-multiply:
5 * 10 = 2 * (x+2)^2
50 = 2(x+2)^2
25 = (x+2)^2
Taking the square root of both sides:
±√25 = ±√[(x+2)^2]
±5 = ±(x+2)
Now we solve for x separately for each sign:
1) +5 = x+2
x = 5 - 2
x = 3
2) -5 = x+2
x = -5 - 2
x = -7
So, we have two possible x-values, x = 3 and x = -7.
To find the corresponding y-coordinate, we substitute each x-value back into the original equation:
For x = 3:
f(x) = 5(3)/(3+2)
= 15/5
= 3
For x = -7:
f(x) = 5(-7)/(-7+2)
= -35/(-5)
= 7
Therefore, the points on the graph of f(x)=5x/(x+2) where the slope of the tangent is 2/5 are (3, 3) and (-7, 7).
So, the correct answer is c) (3,3) and (-7,7).