Find all points on the graph of the function f(x)= 2sin(x)+sin^2(x) at which the tangent line is horizontal.
To find the points where the tangent line is horizontal, we need to find the derivative of the function and set it equal to zero.
Taking the derivative of the function f(x) with respect to x using the chain rule, we have:
f'(x) = 2cos(x) + 2sin(x) * cos(x)
Setting f'(x) equal to zero, we have:
2cos(x) + 2sin(x) * cos(x) = 0
Dividing both sides by 2cos(x), we get:
1 + sin(x) = 0
Solving for sin(x), we have:
sin(x) = -1
Since sin(x) = -1 at x = -π/2, we have one solution.
Now, to find all the points on the graph of the function where the tangent line is horizontal, we substitute x = -π/2 back into the original function:
f(-π/2) = 2sin(-π/2) + sin^2(-π/2)
= -2 + (-1)^2
= -2 + 1
= -1
Hence, the point on the graph of the function where the tangent line is horizontal is (-π/2, -1).