Question

The number of solutions of x^2 = 5 cos(3x+1) in the interval is[-2 \pi , 2 \pi ]

Answer 1

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.