For what value of x will the function f(x) = -3(x - 10)(x - 4) have a maximum value? Find the maximum value.?
this is just a parabola. Its axis of symmetry is midway between the roots.
Naturally, the vertex (maximum y) lies on that axis of symmetry.
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F(x) = -3(x-10)(x-4) = -3(x^2-4x-10x+40) = -3(x^2-14x+40),
F(x) = -3x^2+42x-120.
h = Xv = -B/2A = -42/-6 = 7.
k = Yv = -3*7^2 + 42*7 - 120 = 27.
V(h, k) = V(7, 27). = max point.
So X = 7 at max. point.
To find the value of x for which the function f(x) will have a maximum value, we can start by analyzing the quadratic equation within the function. The quadratic equation is in the form ax^2 + bx + c, where in our case, a = -3, b = 42, and c = -160.
To find the vertex of the quadratic equation, we can use the formula x = -b / (2a). Substituting the values, we get:
x = -42 / (2 * -3)
x = 42 / 6
x = 7
So, the function f(x) will have a maximum value at x = 7.
To find the maximum value, we can substitute the value of x = 7 into the equation f(x). Calculating that gives us:
f(7) = -3(7 - 10)(7 - 4)
= -3(-3)(3)
= -27
Therefore, the maximum value of the function f(x) is -27.