the proportion of the wind's energy that can be obtained from a windmill is given e(x)=4x^3-8x^2+4x where x is slowed find proportion x that max interval (0,1)
To find the maximum proportion of wind energy that can be obtained from a windmill within the interval (0, 1), we need to find the critical points of the given function e(x) = 4x^3 - 8x^2 + 4x and determine which one lies within the interval.
Here's how you can do it step by step:
1. Take the derivative of the function e(x) with respect to x. Let's call this derivative function e'(x).
e'(x) = d/dx (4x^3 - 8x^2 + 4x)
To find e'(x), apply the power rule of differentiation to each term:
e'(x) = 12x^2 - 16x + 4
2. Set the derivative e'(x) equal to zero and solve for x to find the critical points:
12x^2 - 16x + 4 = 0
3. To solve the quadratic equation, you can either factor it or use the quadratic formula. In this case, factoring is not straightforward, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 12, b = -16, and c = 4.
x = (-(-16) ± √((-16)^2 - 4 * 12 * 4)) / (2 * 12)
x = (16 ± √(256 - 192)) / 24
x = (16 ± √64) / 24
x = (16 ± 8) / 24
This gives two possible solutions:
x1 = 24 / 24 = 1
x2 = 8 / 24 = 1/3
4. Determine which critical point lies within the interval (0, 1). In this case, x2 = 1/3, which satisfies the condition.
So, the critical point x = 1/3 corresponds to the maximum proportion of wind energy that can be obtained from a windmill within the interval (0, 1).