Determine coordinates of the local extreme values on a function see for question below

Question

7. Determine the x and y coordinates of the local extreme values on the function 𝑦 = 𝑒^𝑥𝑠𝑖𝑛𝑥 over the

interval 0 ≤ 𝑥 ≤ 2𝜋. Calculate the y coordinates to 3 decimal places. You do not need to indicate

whether the extrema are maxima or minima.

Answer 1

Differentiate the function

f(x) = e^xsin(x)

f'(x) = e^xsin(x)+e^xcos(x) = e^x[sin(x)+cos(x)]

Determine critical values

0 = e^x[sin(x)+cos(x)]

0 = sin(x)+cos(x)

-cos(x) = sin(x)

-1 = tan(x)

x = 3π/4, 7π/4

Hence, the y-coordinates for the local extreme values are:

f(3π/4) = e^3π/4sin(3π/4) = e^3π/4(√2/2) ≈ 7.460 <-- Local Maximum

f(7π/4) = e^3π/4sin(7π/4) = e^7π/4(-√2/2) ≈ -172.641 <-- Local Minimum

As ordered pairs, these are (3π/4,7.460) and (7π/4,-172.641)

Hope this helped!

Answer 2

y' = e^xsinx + e^xcosx = 0; sinx + cosx = 0; tanx = - 1; x = 3π/4 or x = 7π/4.

Points of extrima: (3π/4, e^3π/4sin3π/4) = (3π/4, 7.460); (7π/4, e^7π/4sin7π/4) = (7π/4, - 172.641)