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Find the minimum value of the function f(x)=2x^(2)-18 x+44 to the nearest hundredth. Answer:

Find the minimum value of the function f(x)=2x2−18x+44 f(x)=2 x^{2}-18 x+44 to the nearest hundredth.\newlineAnswer:

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Full solution

Q. Find the minimum value of the function f(x)=2x2−18x+44 f(x)=2 x^{2}-18 x+44 to the nearest hundredth.\newlineAnswer:
  1. Calculate Vertex: To find the minimum value of the quadratic function f(x)=2x2−18x+44f(x) = 2x^2 - 18x + 44, 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} and kk is the value of the function at x=hx = h.
  2. Calculate h: 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=−18b = -18.h=−(−18)/(2⋅2)=184=4.5h = -(-18)/(2\cdot2) = \frac{18}{4} = 4.5
  3. Calculate kk: Next, we calculate the y-coordinate of the vertex, kk, by substituting x=hx = h into the function f(x)f(x).k=f(4.5)=2(4.5)2−18(4.5)+44k = f(4.5) = 2(4.5)^2 - 18(4.5) + 44
  4. Perform Calculation for kk: Now, we perform the calculation for kk.k=2(20.25)−81+44k = 2(20.25) - 81 + 44k=40.5−81+44k = 40.5 - 81 + 44k=−40.5+44k = -40.5 + 44k=3.5k = 3.5
  5. Determine Minimum Value: Since the coefficient of x2x^2 is positive, the parabola opens upwards, and the vertex represents the minimum point of the function. Therefore, the minimum value of the function is kk, which is 3.53.5.

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