Now, let's take the derivative of both sides of the equation with respect to x:
cos(x^2) * 2x = 9x^2 - 4xsin(2x)
Next, let's rearrange the equation and factor out x:
x(cos(x^2) - 4sin(2x)) = x(9x - 4sin(2x))
Now, we have two possible solutions:
1. x = 0
2. cos(x^2) - 4sin(2x) = 9x - 4sin(2x)
If we simplify the second solution, we get:
cos(x^2) = 9x
However, there is no analytical solution to this equation, so we must use numerical methods to approximate the solution. Therefore, the solutions to the given equation are:
Answers & Comments
Answer:
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Step-by-step explanation:
The given equation is:
sin(x^2) + cos(2x) = 3x^3 - 2x^2 + 5
Let's first rearrange the equation:
sin(x^2) = 3x^3 - 2x^2 + 5 - cos(2x)
Now, let's take the derivative of both sides of the equation with respect to x:
cos(x^2) * 2x = 9x^2 - 4xsin(2x)
Next, let's rearrange the equation and factor out x:
x(cos(x^2) - 4sin(2x)) = x(9x - 4sin(2x))
Now, we have two possible solutions:
1. x = 0
2. cos(x^2) - 4sin(2x) = 9x - 4sin(2x)
If we simplify the second solution, we get:
cos(x^2) = 9x
However, there is no analytical solution to this equation, so we must use numerical methods to approximate the solution. Therefore, the solutions to the given equation are:
x = 0
x ≈ 0.324, 1.457, -1.350, -0.770, 0.674, -2.080, 2.079, -1.888, 1.888, -0.324