Find the minimum value of the function $f(x)=0.9 x^{2}+5.6 x+16$ to the nearest hundredth. $\newline$ Answer:

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Q. Find the minimum value of the function

f(x)=0.9 x^{2}+5.6 x+16

to the nearest hundredth.

\newline

Answer:

Identify Coefficients: To find the minimum value of the quadratic function $f(x) = 0.9x^2 + 5.6x + 16$ , 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 $a$ is the coefficient of $x^2$ and $b$ is the coefficient of $x$ .
Calculate x-coordinate: First, we identify the coefficients $a$ and $b$ from the function $f(x) = 0.9x^2 + 5.6x + 16$ . Here, $a = 0.9$ and $b = 5.6$ .
Find y-coordinate: Next, we calculate the x-coordinate of the vertex using the formula $-\frac{b}{2a}$ . Plugging in the values, we get $-\frac{5.6}{2\times0.9}$ .
Substitute $x$ into function: Performing the calculation, we get $-\frac{5.6}{1.8}$ , which equals $-3.111111\ldots$
Calculate minimum value: Now that we have the $x$ -coordinate of the vertex, we can find the $y$ -coordinate, which is the minimum value of the function, by plugging the $x$ -coordinate back into the original function $f(x)$ .
Round to nearest hundredth: We substitute $x = -3.111111...$ into the function $f(x) = 0.9x^2 + 5.6x + 16$ to get $f(-3.111111...) = 0.9(-3.111111...)^2 + 5.6(-3.111111...) + 16$ .
Round to nearest hundredth: We substitute $x = -3.111111...$ into the function $f(x) = 0.9x^2 + 5.6x + 16$ to get $f(-3.111111...) = 0.9(-3.111111...)^2 + 5.6(-3.111111...) + 16$ . Calculating the value, we get $f(-3.111111...) = 0.9(9.679012...) - 17.422222... + 16$ .
Round to nearest hundredth: We substitute $x = -3.111111...$ into the function $f(x) = 0.9x^2 + 5.6x + 16$ to get $f(-3.111111...) = 0.9(-3.111111...)^2 + 5.6(-3.111111...) + 16$ . Calculating the value, we get $f(-3.111111...) = 0.9(9.679012...) - 17.422222... + 16$ . Further simplifying, we get $f(-3.111111...) = 8.711111... - 17.422222... + 16$ .
Round to nearest hundredth: We substitute $x = -3.111111\ldots$ into the function $f(x) = 0.9x^2 + 5.6x + 16$ to get $f(-3.111111\ldots) = 0.9(-3.111111\ldots)^2 + 5.6(-3.111111\ldots) + 16$ . Calculating the value, we get $f(-3.111111\ldots) = 0.9(9.679012\ldots) - 17.422222\ldots + 16$ . Further simplifying, we get $f(-3.111111\ldots) = 8.711111\ldots - 17.422222\ldots + 16$ . Finally, we get $f(-3.111111\ldots) = 7.288889\ldots$
Round to nearest hundredth: We substitute $x = -3.111111\ldots$ into the function $f(x) = 0.9x^2 + 5.6x + 16$ to get $f(-3.111111\ldots) = 0.9(-3.111111\ldots)^2 + 5.6(-3.111111\ldots) + 16$ . Calculating the value, we get $f(-3.111111\ldots) = 0.9(9.679012\ldots) - 17.422222\ldots + 16$ . Further simplifying, we get $f(-3.111111\ldots) = 8.711111\ldots - 17.422222\ldots + 16$ . Finally, we get $f(-3.111111\ldots) = 7.288889\ldots$ Rounding to the nearest hundredth, the minimum value of the function is approximately $7.29$ .