Critical Points problem of Calculus

Question

Find the critical points and determine the type of critical point:

y = e^x + sinx

Answer 1

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y = e^x + sin x

y' = e^x + cos x

For x≧0, we should note that e^x is always greater than cos x, implying y is monotonic , i.e. y has no critical point for x≧0

For x<0, 0 < e^x < 1, monotonic and -1 < cos x < 1, periodic. Thus y' cuts through x-axis occasionally. So all the solutions for x to the equation e^x + cos x = 0 are the critical points. They are all stationary points.

(In fact, e^x is so small that can be neglected and y' ≒ cos x. A close estimation to the critical points are x ≒ -pi/2, -3pi/2, -5pi/2 ......)

Hope the above information helps =)

By 小儒