Find the minimum value of the function $f(x)=1.7 x^{2}+24.1 x+89$ to the nearest hundredth. $\newline$ Answer:

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Q. Find the minimum value of the function

f(x)=1.7 x^{2}+24.1 x+89

to the nearest hundredth.

\newline

Answer:

Find Vertex Coefficients: To find the minimum value of the quadratic function $f(x) = 1.7x^2 + 24.1x + 89$ , we need to find the vertex of the parabola. The $x$ -coordinate of the vertex can be found using the formula $-\frac{b}{2a}$ , where $a$ and $b$ are the coefficients from the quadratic equation $ax^2 + bx + c$ .
Calculate X-coordinate: First, we identify the coefficients $a$ and $b$ from the function $f(x) = 1.7x^2 + 24.1x + 89$ . Here, $a = 1.7$ and $b = 24.1$ .
Substitute $X$ into Function: Next, we apply the formula to find the $x$ -coordinate of the vertex: $x = -\frac{b}{2a} = -\frac{24.1}{2 \times 1.7}$ .
Calculate Minimum Value: Calculating the $x$ -coordinate gives us $x = -24.1 / 3.4 = -7.088235294117647$ .
Calculate Minimum Value: Calculating the x-coordinate gives us $x = \frac{-24.1}{3.4} = -7.088235294117647$ .Now that we have the x-coordinate of the vertex, we can find the minimum value of the function by substituting $x$ back into the original equation $f(x) = 1.7x^2 + 24.1x + 89$ .
Calculate Minimum Value: Calculating the x-coordinate gives us $x = \frac{-24.1}{3.4} = -7.088235294117647$ .Now that we have the x-coordinate of the vertex, we can find the minimum value of the function by substituting $x$ back into the original equation $f(x) = 1.7x^2 + 24.1x + 89$ .Substituting $x = -7.088235294117647$ into the function gives us $f(-7.088235294117647) = 1.7(-7.088235294117647)^2 + 24.1(-7.088235294117647) + 89$ .
Calculate Minimum Value: Calculating the x-coordinate gives us $x = \frac{-24.1}{3.4} = -7.088235294117647$ .Now that we have the x-coordinate of the vertex, we can find the minimum value of the function by substituting $x$ back into the original equation $f(x) = 1.7x^2 + 24.1x + 89$ .Substituting $x = -7.088235294117647$ into the function gives us $f(-7.088235294117647) = 1.7(-7.088235294117647)^2 + 24.1(-7.088235294117647) + 89$ .Performing the calculations, we get $f(-7.088235294117647) = 1.7(50.244) - 170.825 + 89$ .
Calculate Minimum Value: Calculating the x-coordinate gives us $x = \frac{-24.1}{3.4} = -7.088235294117647$ .Now that we have the x-coordinate of the vertex, we can find the minimum value of the function by substituting $x$ back into the original equation $f(x) = 1.7x^2 + 24.1x + 89$ .Substituting $x = -7.088235294117647$ into the function gives us $f(-7.088235294117647) = 1.7(-7.088235294117647)^2 + 24.1(-7.088235294117647) + 89$ .Performing the calculations, we get $f(-7.088235294117647) = 1.7(50.244) - 170.825 + 89$ .Further simplifying, we get $f(-7.088235294117647) = 85.4148 - 170.825 + 89$ .
Calculate Minimum Value: Calculating the x-coordinate gives us $x = \frac{-24.1}{3.4} = -7.088235294117647$ .Now that we have the x-coordinate of the vertex, we can find the minimum value of the function by substituting $x$ back into the original equation $f(x) = 1.7x^2 + 24.1x + 89$ .Substituting $x = -7.088235294117647$ into the function gives us $f(-7.088235294117647) = 1.7(-7.088235294117647)^2 + 24.1(-7.088235294117647) + 89$ .Performing the calculations, we get $f(-7.088235294117647) = 1.7(50.244) - 170.825 + 89$ .Further simplifying, we get $f(-7.088235294117647) = 85.4148 - 170.825 + 89$ .Finally, we add the numbers to get the minimum value: $f(-7.088235294117647) = 85.4148 - 170.825 + 89 = 3.5898$ .
Calculate Minimum Value: Calculating the x-coordinate gives us $x = \frac{-24.1}{3.4} = -7.088235294117647$ .Now that we have the x-coordinate of the vertex, we can find the minimum value of the function by substituting $x$ back into the original equation $f(x) = 1.7x^2 + 24.1x + 89$ .Substituting $x = -7.088235294117647$ into the function gives us $f(-7.088235294117647) = 1.7(-7.088235294117647)^2 + 24.1(-7.088235294117647) + 89$ .Performing the calculations, we get $f(-7.088235294117647) = 1.7(50.244) - 170.825 + 89$ .Further simplifying, we get $f(-7.088235294117647) = 85.4148 - 170.825 + 89$ .Finally, we add the numbers to get the minimum value: $f(-7.088235294117647) = 85.4148 - 170.825 + 89 = 3.5898$ .Rounding to the nearest hundredth, the minimum value of the function is approximately $3.59$ .