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Find the maximum value of the function f(x)=-1.6x^(2)+10 x-10.8 to the nearest hundredth. Answer:

Find the maximum value of the function f(x)=−1.6x2+10x−10.8 f(x)=-1.6 x^{2}+10 x-10.8 to the nearest hundredth.\newlineAnswer:

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Full solution

Q. Find the maximum value of the function f(x)=−1.6x2+10x−10.8 f(x)=-1.6 x^{2}+10 x-10.8 to the nearest hundredth.\newlineAnswer:
  1. Find Vertex: To find the maximum value of the quadratic function f(x)=−1.6x2+10x−10.8f(x) = -1.6x^2 + 10x - 10.8, we need to find the vertex of the parabola. Since the coefficient of x2x^2 is negative (−1.6-1.6), 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=10b = 10.\newlinex-coordinate of vertex = −b2a=−102×−1.6=−10−3.2=3.125-\frac{b}{2a} = -\frac{10}{2 \times -1.6} = -\frac{10}{-3.2} = 3.125
  3. Substitute xx into function: Next, we substitute x=3.125x = 3.125 into the function to find the yy-coordinate of the vertex, which will give us the maximum value of the function.\newlinef(3.125)=−1.6(3.125)2+10(3.125)−10.8f(3.125) = -1.6(3.125)^2 + 10(3.125) - 10.8
  4. Perform calculations: Now we perform the calculations:\newlinef(3.125)=−1.6(9.765625)+31.25−10.8f(3.125) = -1.6(9.765625) + 31.25 - 10.8\newlinef(3.125)=−15.625+31.25−10.8f(3.125) = -15.625 + 31.25 - 10.8\newlinef(3.125)=15.625−10.8f(3.125) = 15.625 - 10.8\newlinef(3.125)=4.825f(3.125) = 4.825
  5. Round to nearest hundredth: Finally, we round the result to the nearest hundredth to get the maximum value of the function.\newlineMaximum value of f(x)≈4.83f(x) \approx 4.83

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