Answer:
y = 3x - 2 (smaller y-intercept)
y = 3x + 2 (larger y-intercept)
Step-by-step explanation:
First let's write the generic equation of a line:
y = ax + b
This line needs to be parallel to the line 3x - y + 5 = 0, so it needs to have the same slope of this line.
The line 3x - y + 5 = 0 has a slope of 3, so our line has a = 3:
y = 3x + b
Now we need to find the values of b that make this line tangent to the function f(x) = x^3
Let's first find the derivative of f(x) in relation to x:
df(x)/dx = 3x^2
This derivative is the slope of the tangent line to the function for any value of x. We need a slope of 3, so:
3x^2 = 3
x^2 = 1
x = ±1
Now, to find the y-values, we have:
f(1) = 1^3 = 1
f(-1) = (-1)^3 = -1
So, using the points (1,1) and (-1,-1) in our parallel line, we have:
first line using (1,1) : 1 = 3*1 + b
b = -2
second line using (-1,-1) : -1 = -3*1 + b
b = 2
The value of b is the y-intercept of the line, so the line with smaller y-intercept is y = 3x - 2, and the line with larger y-intercept is y = 3x + 2
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