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Find the minimum value of the function f(x)=0.7x^(2)-7x+22.1 to the nearest hundredth. Answer:

Find the minimum value of the function f(x)=0.7x2−7x+22.1 f(x)=0.7 x^{2}-7 x+22.1 to the nearest hundredth.\newlineAnswer:

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Q. Find the minimum value of the function f(x)=0.7x2−7x+22.1 f(x)=0.7 x^{2}-7 x+22.1 to the nearest hundredth.\newlineAnswer:
  1. Calculate Vertex: To find the minimum value of the quadratic function f(x)=0.7x2−7x+22.1f(x) = 0.7x^2 - 7x + 22.1, 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 (0.70.7), the parabola opens upwards, and thus the vertex represents the minimum point.
  2. Find x-coordinate: First, we calculate the x-coordinate of the vertex, hh, using the formula h=−b/(2a)h = -b/(2a). Here, a=0.7a = 0.7 and b=−7b = -7.\newlineh=−(−7)/(2×0.7)=7/1.4=5h = -(-7) / (2 \times 0.7) = 7 / 1.4 = 5.
  3. Find y-coordinate: Next, we substitute x=5x = 5 into the function to find the y-coordinate of the vertex, kk, which will give us the minimum value of the function.f(5)=0.7(5)2−7(5)+22.1=0.7(25)−35+22.1=17.5−35+22.1=−17.5+22.1=4.6.f(5) = 0.7(5)^2 - 7(5) + 22.1 = 0.7(25) - 35 + 22.1 = 17.5 - 35 + 22.1 = -17.5 + 22.1 = 4.6.
  4. Determine Minimum Value: Now we have the vertex of the parabola, which is at the point (5,4.6)(5, 4.6). The yy-coordinate of the vertex, kk, is the minimum value of the function.

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