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Find the maximum value of the function f(x)=-1.6x^(2)+20.1 x-71 to the nearest hundredth. Answer:

Find the maximum value of the function f(x)=−1.6x2+20.1x−71 f(x)=-1.6 x^{2}+20.1 x-71 to the nearest hundredth.\newlineAnswer:

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Full solution

Q. Find the maximum value of the function f(x)=−1.6x2+20.1x−71 f(x)=-1.6 x^{2}+20.1 x-71 to the nearest hundredth.\newlineAnswer:
  1. Find Vertex: To find the maximum value of the quadratic function f(x)=−1.6x2+20.1x−71f(x) = -1.6x^2 + 20.1x - 71, we need to find the vertex of the parabola. Since the coefficient of x2x^2 is negative, the parabola opens downwards, and the vertex will give us the maximum value. The xx-coordinate of the vertex can be found using the formula −b2a-\frac{b}{2a}, where aa is the coefficient of x2x^2 and bb is the coefficient of xx.
  2. Calculate x-coordinate: First, we calculate the x-coordinate of the vertex using the formula −b2a-\frac{b}{2a}. Here, a=−1.6a = -1.6 and b=20.1b = 20.1.
    x-coordinate of vertex = −b2a-\frac{b}{2a} = −20.12×−1.6-\frac{20.1}{2 \times -1.6} = −20.1−3.2-\frac{20.1}{-3.2} = 66.2812528125
  3. Calculate y-coordinate: Now that we have the x-coordinate of the vertex, we can find the y-coordinate, which is the maximum value of the function, by substituting x=6.28125x = 6.28125 into the function f(x)f(x).f(6.28125)=−1.6(6.28125)2+20.1(6.28125)−71f(6.28125) = -1.6(6.28125)^2 + 20.1(6.28125) - 71
  4. Perform Calculations: Let's perform the calculations:\newlinef(6.28125)=−1.6(39.452515625)+126.25−71f(6.28125) = -1.6(39.452515625) + 126.25 - 71\newlinef(6.28125)=−63.123225+126.25−71f(6.28125) = -63.123225 + 126.25 - 71\newlinef(6.28125)=63.126775f(6.28125) = 63.126775
  5. Recalculate f(6.28125)f(6.28125): We will redo the calculation for f(6.28125)f(6.28125) correctly: f(6.28125)=−1.6(6.28125)2+20.1(6.28125)−71f(6.28125) = -1.6(6.28125)^2 + 20.1(6.28125) - 71 f(6.28125)=−1.6×39.452515625+126.2503125−71f(6.28125) = -1.6 \times 39.452515625 + 126.2503125 - 71 f(6.28125)=−63.123225+126.2503125−71f(6.28125) = -63.123225 + 126.2503125 - 71 f(6.28125)=63.1270875−71f(6.28125) = 63.1270875 - 71 f(6.28125)=−7.8729125f(6.28125) = -7.8729125 Since we are looking for the maximum value, we need to round this to the nearest hundredth.
  6. Round to Nearest Hundredth: The maximum value of the function, rounded to the nearest hundredth, is −7.87-7.87.

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