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Find the minimum value of the function f(x)=x^(2)-2.9 x+10 to the nearest hundredth. Answer:

Find the minimum value of the function f(x)=x2−2.9x+10 f(x)=x^{2}-2.9 x+10 to the nearest hundredth.\newlineAnswer:

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Full solution

Q. Find the minimum value of the function f(x)=x2−2.9x+10 f(x)=x^{2}-2.9 x+10 to the nearest hundredth.\newlineAnswer:
  1. Calculate Vertex: To find the minimum value of the quadratic function f(x)=x2−2.9x+10f(x) = x^2 - 2.9x + 10, 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 x-coordinate: First, we calculate the x-coordinate of the vertex, hh. For the given function, a=1a = 1 and b=−2.9b = -2.9. So, h=−(−2.9)/(2⋅1)=2.9/2=1.45h = -(-2.9)/(2\cdot1) = 2.9/2 = 1.45.
  3. Calculate y-coordinate: Next, we calculate the y-coordinate of the vertex, kk, by substituting x=hx = h into the function. So, k=f(1.45)=(1.45)2−2.9(1.45)+10k = f(1.45) = (1.45)^2 - 2.9(1.45) + 10.
  4. Perform Calculation: Now, we perform the calculation: k=(1.45)2−2.9(1.45)+10=2.1025−4.205+10=7.8975k = (1.45)^2 - 2.9(1.45) + 10 = 2.1025 - 4.205 + 10 = 7.8975.
  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 we have calculated to be 7.89757.8975.
  6. Round to Nearest Hundredth: Finally, we round kk to the nearest hundredth to get the minimum value of the function. The rounded value is 7.907.90.

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