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Find the minimum value of the function f(x)=2x^(2)-13.3 x+16.2 to the nearest hundredth. Answer:

Find the minimum value of the function f(x)=2x2−13.3x+16.2 f(x)=2 x^{2}-13.3 x+16.2 to the nearest hundredth.\newlineAnswer:

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Full solution

Q. Find the minimum value of the function f(x)=2x2−13.3x+16.2 f(x)=2 x^{2}-13.3 x+16.2 to the nearest hundredth.\newlineAnswer:
  1. Calculate Vertex: To find the minimum value of the quadratic function f(x)=2x2−13.3x+16.2f(x) = 2x^2 - 13.3x + 16.2, 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}. Here, a=2a = 2 and b=−13.3b = -13.3. \newlineh=−(−13.3)/(2×2)=13.34=3.325h = -(-13.3) / (2 \times 2) = \frac{13.3}{4} = 3.325
  3. Substitute xx into function: Next, we substitute x=hx = h into the function to find the y-coordinate of the vertex, kk, which will give us the minimum value of the function.\newlinef(3.325)=2(3.325)2−13.3(3.325)+16.2f(3.325) = 2(3.325)^2 - 13.3(3.325) + 16.2
  4. Perform calculations: Now we perform the calculations:\newlinef(3.325)=2(11.075625)−44.2425+16.2f(3.325) = 2(11.075625) - 44.2425 + 16.2\newlinef(3.325)=22.15125−44.2425+16.2f(3.325) = 22.15125 - 44.2425 + 16.2
  5. Simplify expression: Simplify the expression to find the value of kk:f(3.325)=−22.09125+16.2f(3.325) = -22.09125 + 16.2f(3.325)=−5.89125f(3.325) = -5.89125
  6. Round to nearest hundredth: Round the result to the nearest hundredth:\newlineThe minimum value of the function f(x)f(x) to the nearest hundredth is −5.89-5.89.

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