minimum value of the function \( f(x)=2 x^{2}-7 x+10.8 \) to the

To find the minimum value of the function \( f(x)=2x^{2}-7x+10.8 \), we need to find the vertex of the parabola represented by the function. The vertex of a parabola in the form \( f(x)=ax^{2}+bx+c \) is given by the formula \( x=-\frac{b}{2a} \). In this case, the function is \( f(x)=2x^{2}-7x+10.8 \), so \( a=2 \) and \( b=-7 \). Let's calculate the x-coordinate of the vertex using the formula \( x=-\frac{b}{2a} \). Calculate the value by following steps: - step0: Calculate: \(-\frac{\left(-7\right)}{\left(2\times 2\right)}\) - step1: Remove the parentheses: \(-\frac{-7}{2\times 2}\) - step2: Use the rules for multiplication and division: \(\frac{7}{2\times 2}\) - step3: Multiply the numbers: \(\frac{7}{4}\) The x-coordinate of the vertex is \( x=\frac{7}{4} \) or \( x=1.75 \). Now, we can substitute this value back into the function \( f(x)=2x^{2}-7x+10.8 \) to find the minimum value of the function. Let's calculate this. Calculate the value by following steps: - step0: Calculate: \(2\left(\frac{7}{4}\right)^{2}-7\times \frac{7}{4}+10.8\) - step1: Multiply the terms: \(\frac{49}{8}-7\times \frac{7}{4}+10.8\) - step2: Multiply the numbers: \(\frac{49}{8}-\frac{49}{4}+10.8\) - step3: Rewrite the expression: \(\frac{49}{8}-\frac{49}{4}+\frac{54}{5}\) - step4: Reduce fractions to a common denominator: \(\frac{49\times 5}{8\times 5}-\frac{49\times 10}{4\times 10}+\frac{54\times 8}{5\times 8}\) - step5: Calculate: \(\frac{245}{40}-\frac{490}{40}+\frac{432}{40}\) - step6: Transform the expression: \(\frac{245-490+432}{40}\) - step7: Calculate: \(\frac{187}{40}\) The minimum value of the function \( f(x)=2x^{2}-7x+10.8 \) is \( 4.675 \).