Answer :
The angles of intersection between the curves f(x) = x2 and g(x) = sqrt(x) are x = 0 and x = 1. This solves to be 4.0 degrees, which is not one of the options.
To find the angles of intersection between the curves f(x) = x^2 and g(x) = √(x), we need to set the two functions equal to each other and solve for x.
Setting the functions equal to each other:
[tex]x^2 = √(x)[/tex]
To solve for x, we'll square both sides of the equation:
[tex](x^2)^2 = (√(x))^2x^4 = x[/tex]
Now, we can rearrange the equation and factor out x:
[tex]x^4 - x = 0x(x^3 - 1) = 0[/tex]
From this equation, we can see that x = 0 is one solution. To find the other solutions, we set x^3 - 1 = 0 and solve for x:
[tex]x^3 - 1 = 0(x - 1)(x^2 + x + 1) = 0[/tex]
This gives us two more solutions: x = 1 and the solutions of the quadratic equation [tex]x^2 + x + 1 = 0[/tex].
Let's solve the quadratic equation using the quadratic formula:
[tex]x = (-b ± √(b^2 - 4ac)) / (2a)[/tex]
For this equation, a = 1, b = 1, and c = 1. Plugging these values into the quadratic formula:
[tex]x = (-1 ± √(1^2 - 4(1)(1))) / (2(1))x = (-1 ± √(1 - 4)) / 2x = (-1 ± √(-3)) / 2[/tex]
Since the square root of a negative number is not a real number, there are no real solutions for the quadratic equation [tex]x^2 + x + 1 = 0[/tex].
Therefore, the angles of intersection between the curves [tex]f(x) = x^2[/tex] and
[tex]g(x) = √(x)[/tex] are-
x = 0 and x = 1.
So, the correct answer from the multiple-choice options is 4. 0 Degrees.
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