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Find the minimum value of the function f(x)=x^(2)-2.7 x+5 to the nearest hundredth. Answer:

Find the minimum value of the function f(x)=x2−2.7x+5 f(x)=x^{2}-2.7 x+5 to the nearest hundredth.\newlineAnswer:

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Full solution

Q. Find the minimum value of the function f(x)=x2−2.7x+5 f(x)=x^{2}-2.7 x+5 to the nearest hundredth.\newlineAnswer:
  1. Calculate Vertex: To find the minimum value of the quadratic function f(x)=x2−2.7x+5f(x) = x^2 - 2.7x + 5, 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 k=f(h)k = f(h).
  2. Find x-coordinate: First, we calculate the x-coordinate of the vertex, hh. For the given function, a=1a = 1 and b=−2.7b = -2.7. So, h=−(−2.7)/(2⋅1)=2.7/2=1.35h = -(-2.7)/(2\cdot1) = 2.7/2 = 1.35.
  3. Find y-coordinate: Next, we calculate the y-coordinate of the vertex, kk, by evaluating the function at x=hx = h. So, k=f(1.35)=(1.35)2−2.7(1.35)+5k = f(1.35) = (1.35)^2 - 2.7(1.35) + 5.
  4. Perform Calculation: Now, we perform the calculation: k=(1.35)2−2.7(1.35)+5=1.8225−3.645+5=3.1775k = (1.35)^2 - 2.7(1.35) + 5 = 1.8225 - 3.645 + 5 = 3.1775.
  5. Determine Parabola Direction: Since the coefficient of the x2x^2 term is positive, the parabola opens upwards, and the vertex represents the minimum point of the function.
  6. Round Minimum Value: Finally, we round the minimum value to the nearest hundredth. The minimum value of the function is approximately 3.183.18.

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