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Find the minimum value of the function f(x)=0.3x^(2)-1.2 x+5 to the nearest hundredth. Answer:

Find the minimum value of the function f(x)=0.3x2−1.2x+5 f(x)=0.3 x^{2}-1.2 x+5 to the nearest hundredth.\newlineAnswer:

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Q. Find the minimum value of the function f(x)=0.3x2−1.2x+5 f(x)=0.3 x^{2}-1.2 x+5 to the nearest hundredth.\newlineAnswer:
  1. Calculate Vertex: To find the minimum value of the quadratic function f(x)=0.3x2−1.2x+5f(x) = 0.3x^2 - 1.2x + 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 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, using the formula h=−b2ah = -\frac{b}{2a}. Here, a=0.3a = 0.3 and b=−1.2b = -1.2.h=−(−1.2)/(2×0.3)=1.20.6=2h = -(-1.2) / (2 \times 0.3) = \frac{1.2}{0.6} = 2.
  3. Calculate y-coordinate: Next, we calculate the y-coordinate of the vertex, kk, by substituting x=hx = h into the function f(x)f(x). \newlinek=f(2)=0.3(2)2−1.2(2)+5=0.3(4)−2.4+5=1.2−2.4+5k = f(2) = 0.3(2)^2 - 1.2(2) + 5 = 0.3(4) - 2.4 + 5 = 1.2 - 2.4 + 5.
  4. Perform Calculation for k: Now, we perform the calculation for k. k=1.2−2.4+5=−1.2+5=3.8k = 1.2 - 2.4 + 5 = -1.2 + 5 = 3.8.
  5. Determine Parabola Direction: Since the coefficient of x2x^2 is positive (a = 0.3 > 0), the parabola opens upwards, and the vertex represents the minimum point of the function.
  6. Round to Nearest Hundredth: Finally, we round the minimum value kk to the nearest hundredth.\newlineThe minimum value of the function to the nearest hundredth is 3.803.80.

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