python Find the maximum of the function sin^2(x-2)e^-1
时间: 2024-06-07 16:05:56 浏览: 24
To find the maximum of the function sin^2(x-2)e^-1, we need to take its derivative and set it equal to zero:
f(x) = sin^2(x-2)e^-1
f'(x) = 2sin(x-2)cos(x-2)e^-1 - sin^2(x-2)e^-1
Setting f'(x) equal to zero and solving for x, we get:
2sin(x-2)cos(x-2)e^-1 - sin^2(x-2)e^-1 = 0
sin(x-2)[2cos(x-2) - sin(x-2)]e^-1 = 0
Either sin(x-2) = 0 or 2cos(x-2) - sin(x-2) = 0.
If sin(x-2) = 0, then x = 2kπ + π/2 for some integer k.
If 2cos(x-2) - sin(x-2) = 0, then 2cos(x-2) = sin(x-2), which means:
tan(x-2) = 2
x = arctan(2) + 2 ≈ 2.107
Now we need to check which of these values of x gives the maximum of the function. We can do this by plugging each value into the original function and comparing the results:
f(2kπ + π/2) = e^-1
f(arctan(2) + 2) ≈ 0.172
Therefore, the maximum value of the function sin^2(x-2)e^-1 occurs at x ≈ 2.107, where f(x) ≈ 0.172.
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