Final answer:
To find the number of solutions of the equation x^2=5cos(3x+1) in the interval [-2π, 2π], we can set the cosine function equal to the quadratic equation and solve for x. The number of solutions will be the number of valid x-values in the interval.
Step-by-step explanation:
The equation x^2 = 5cos(3x+1) has multiple solutions in the interval [-2π, 2π]. To find the number of solutions, we need to determine the number of times the cosine function crosses the parabola of the quadratic equation. The cosine function crosses the x-axis when its argument, 3x+1, is equal to (2n+1)π/2, where n is an integer.
So, we can set 3x+1 = (2n+1)π/2 and solve for x. The number of solutions will be the number of valid x-values in the interval [-2π, 2π].
Let's solve the equation:
3x+1 = (2n+1)π/2
3x = (2n+1)π/2 - 1
x = ((2n+1)π/2 - 1)/3
Now, we can substitute different values of n and check if the resulting x-values are within the given interval. Count the number of x-values that satisfy the conditions.