Question

question
find the minimum value of the function $f(x) = 1.7x^2 + 25.6x + 89.7$ to the nearest hundredth.

Explanation:

Step1: Identify the formula for vertex of parabola

For a quadratic function \( f(x) = ax^2 + bx + c \), the x - coordinate of the vertex is given by \( x=-\frac{b}{2a} \). Here, \( a = 1.7 \), \( b=25.6 \), \( c = 89.7 \).
First, calculate the x - coordinate of the vertex: \( x=-\frac{25.6}{2\times1.7}=-\frac{25.6}{3.4}\approx - 7.5294 \)

Step2: Substitute x into the function

Now, substitute \( x\approx - 7.5294 \) into the function \( f(x)=1.7x^{2}+25.6x + 89.7 \)
\( f(-7.5294)=1.7\times(-7.5294)^{2}+25.6\times(-7.5294)+89.7 \)
First, calculate \( (-7.5294)^{2}\approx56.691 \)
Then, \( 1.7\times56.691\approx96.375 \)
Next, \( 25.6\times(-7.5294)\approx - 192.753 \)
Now, \( f(-7.5294)=96.375-192.753 + 89.7 \)
\( 96.375+89.7=186.075 \)
\( 186.075-192.753=-6.678\approx - 6.68 \)

Answer:

The minimum value of the function is approximately \(-6.68\)