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Find the minimum value of the function f(x)=x^(2)-4.2 x+12 to the nearest hundredth. Answer:

Find the minimum value of the function f(x)=x2−4.2x+12 f(x)=x^{2}-4.2 x+12 to the nearest hundredth.\newlineAnswer:

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Full solution

Q. Find the minimum value of the function f(x)=x2−4.2x+12 f(x)=x^{2}-4.2 x+12 to the nearest hundredth.\newlineAnswer:
  1. Calculate Vertex: To find the minimum value of the quadratic function f(x)=x2−4.2x+12f(x) = x^2 - 4.2x + 12, 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. Calculate x-coordinate: First, we calculate the x-coordinate of the vertex, hh. For the given function, a=1a = 1 and b=−4.2b = -4.2. Thus, h=−(−4.2)/(2⋅1)=4.2/2=2.1h = -(-4.2)/(2\cdot1) = 4.2/2 = 2.1.
  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(2.1)=(2.1)2−4.2×(2.1)+12k = f(2.1) = (2.1)^2 - 4.2 \times (2.1) + 12.
  4. Perform calculation for k: Now, we perform the calculation for k. k=(2.1)2−4.2∗(2.1)+12=4.41−8.82+12k = (2.1)^2 - 4.2*(2.1) + 12 = 4.41 - 8.82 + 12.
  5. Simplify expression for kk: Simplifying the expression for kk gives us k=4.41−8.82+12=−4.41+12=7.59k = 4.41 - 8.82 + 12 = -4.41 + 12 = 7.59.
  6. 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 7.597.59.

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