a) To find an equation of a line that is tangent to the curve y = 2sin(x) and has the maximum slope, we can start by finding the derivative of the given function. The derivative will give us the slope of the tangent line at any point on the curve.
The derivative of y = 2sin(x) can be found using the chain rule, which states that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x). In this case, g(x) = x and f(x) = 2sin(x).
To find the derivative, we differentiate f(x) = 2sin(x) with respect to x:
f'(x) = d/dx(2sin(x))
= 2 * cos(x)
Now, we have the derivative of y = 2sin(x) as f'(x) = 2cos(x).
To find the maximum slope, we set the derivative equal to zero and solve for x:
2cos(x) = 0
Since cos(x) is equal to zero when x = Ļ/2 + kĻ and x = 3Ļ/2 + kĻ, where k is an integer, we have two potential maximum points.
To find the maximum slope, we can evaluate the derivative at these two points:
f'(Ļ/2) = 2cos(Ļ/2) = 0
f'(3Ļ/2) = 2cos(3Ļ/2) = 0
Both of these points have a slope of zero, which means they are not the maximum slope. However, let's evaluate the slope at a point near these values to determine the maximum slope.
Let's consider the point x = Ļ/4:
f'(Ļ/4) = 2cos(Ļ/4) = ā2
This value of ā2 is the maximum slope.
Now that we have the maximum slope, we need to find the equation of the tangent line that has this slope and passes through a point on the curve y = 2sin(x).
To find the equation of the tangent line, we need a point (x1, y1) on the curve. Let's consider x = Ļ/4:
y = 2sin(Ļ/4) = 2/ā2 = ā2
So, we have the point (Ļ/4, ā2).
The equation of the line can be written as y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.
Plugging in the values, we have:
y - ā2 = ā2(x - Ļ/4)
Simplifying it, we get:
y = ā2x - Ļ/2
Therefore, the equation of the tangent line with the maximum slope is y = ā2x - Ļ/2.
b) No, there is only one solution because we found the maximum slope by evaluating the derivative of y = 2sin(x) and setting it equal to zero. This gave us two potential maximum points, but upon further evaluation, we found that both of them have a slope of zero. The maximum slope of ā2 was obtained at the point x = Ļ/4, and we derived the equation of the tangent line passing through this point and having the maximum slope. Therefore, there is only one solution.