Question

Find the points at which the tangent line is horizontal..

f(x) = cos x + sin x; 0 <=x <= 2x

Answer 1

f(x) = cos x + sin x


f'(x) = -sin x + cos x


Now set the derivative function to zero and solve for x


f'(x) = -sin x + cos x


0 = -sin x + cos x


sin x = cos x


Use the unit circle to see that the solution is x = pi/4


So the tangent line is horizontal at x = pi/4


Note: You can add multiples of pi to the solution for x to find other solutions.

Answer 2

Hi jim_thompson5190,

Thanks for your answer. That's what I got too, but the back of the book gives the following answer.

(pi/4, sqrt(2)), (5pi/4, - sqrt(2))

That has me a bit confused...

Answer 3

pi/4 is the x value and f(pi/4) = sqrt(2) (ie plug in x = pi/4 into the original function and simplify). So one point where the tangent is horizontal is (pi/4, sqrt(2))


Now add pi to pi/4 to get 5pi/4. This is another value of x where the tangent is horizontal. It turns out that f(5pi/4) = -sqrt(2), which means that another point where the the tangent is horizontal is (5pi/4, -sqrt(2))

Answer 4

Thanks so much for your answer!