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Find the minimum value of the function f(x)=2x^(2)-17 x+40.8 to the nearest hundredth. Answer:

Find the minimum value of the function f(x)=2x2−17x+40.8 f(x)=2 x^{2}-17 x+40.8 to the nearest hundredth.\newlineAnswer:

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Q. Find the minimum value of the function f(x)=2x2−17x+40.8 f(x)=2 x^{2}-17 x+40.8 to the nearest hundredth.\newlineAnswer:
  1. Calculate Vertex: To find the minimum value of the quadratic function f(x)=2x2−17x+40.8f(x) = 2x^2 - 17x + 40.8, we can use the vertex formula. The vertex of a parabola given by f(x)=ax2+bx+cf(x) = ax^2 + bx + c is at the point (h,k)(h, k), where h=−b2ah = -\frac{b}{2a}. Since the coefficient of x2x^2 is positive, the parabola opens upwards, and the vertex represents the minimum point.
  2. Find x-coordinate: First, we calculate the x-coordinate of the vertex, hh, using the formula h=−b2ah = -\frac{b}{2a}. In our function, a=2a = 2 and b=−17b = -17.\newlineh=−(−17)/(2×2)=174=4.25h = -(-17) / (2 \times 2) = \frac{17}{4} = 4.25
  3. Substitute xx: Next, we substitute x=hx = h back into the function to find the y-coordinate of the vertex, kk, which will give us the minimum value of the function.\newlinef(4.25)=2(4.25)2−17(4.25)+40.8f(4.25) = 2(4.25)^2 - 17(4.25) + 40.8
  4. Perform Calculations: Now we perform the calculations:\newlinef(4.25)=2(18.0625)−72.25+40.8f(4.25) = 2(18.0625) - 72.25 + 40.8\newlinef(4.25)=36.125−72.25+40.8f(4.25) = 36.125 - 72.25 + 40.8\newlinef(4.25)=−36.125+40.8f(4.25) = -36.125 + 40.8\newlinef(4.25)=4.675f(4.25) = 4.675
  5. Round Minimum Value: Finally, we round the minimum value to the nearest hundredth.\newlineThe minimum value of the function f(x)f(x) to the nearest hundredth is 4.684.68.

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