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Find the minimum value of the function f(x)=0.4x^(2)+x+5 to the nearest hundredth. Answer:

Find the minimum value of the function f(x)=0.4x2+x+5 f(x)=0.4 x^{2}+x+5 to the nearest hundredth.\newlineAnswer:

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Q. Find the minimum value of the function f(x)=0.4x2+x+5 f(x)=0.4 x^{2}+x+5 to the nearest hundredth.\newlineAnswer:
  1. Calculate Vertex: To find the minimum value of the quadratic function f(x)=0.4x2+x+5f(x) = 0.4x^2 + x + 5, we can use the vertex formula for a parabola. 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. Find x-coordinate: First, we calculate the x-coordinate of the vertex, hh, using the formula h=−b/(2a)h = -b/(2a). In our function, a=0.4a = 0.4 and b=1b = 1, so h=−1/(2×0.4)=−1/0.8=−1.25h = -1/(2\times0.4) = -1/0.8 = -1.25.
  3. Find y-coordinate: Next, we calculate the y-coordinate of the vertex, kk, by substituting x=hx = h into the function. So we evaluate f(−1.25)=0.4(−1.25)2+(−1.25)+5f(-1.25) = 0.4(-1.25)^2 + (-1.25) + 5.
  4. Evaluate f(−1.25)f(-1.25): Calculating f(−1.25)f(-1.25) gives us 0.4(1.5625)−1.25+5=0.625−1.25+5=4.3750.4(1.5625) - 1.25 + 5 = 0.625 - 1.25 + 5 = 4.375.
  5. Determine Parabola Direction: Since the coefficient of x2x^2 is positive (a = 0.4 > 0), the parabola opens upwards, and the vertex represents the minimum point of the function.
  6. Calculate Minimum Value: Therefore, the minimum value of the function f(x)f(x) is 4.3754.375, and since we need to round to the nearest hundredth, the minimum value is 4.384.38.

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