Question

question
find the maximum value of the function $f(x) = -1.8x^2 + 14.7x - 27$ 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, \( a=-1.8 \), \( b = 14.7 \), \( c=-27 \).
So, \( x=-\frac{14.7}{2\times(-1.8)}=\frac{14.7}{3.6}\approx4.0833 \)

Step2: Find the maximum value by substituting x into the function

Substitute \( x\approx4.0833 \) into \( f(x)=-1.8x^{2}+14.7x - 27 \)
\( f(4.0833)=-1.8\times(4.0833)^{2}+14.7\times4.0833-27 \)
First, calculate \( (4.0833)^{2}\approx16.6739 \)
Then, \( -1.8\times16.6739\approx - 30.0130 \)
\( 14.7\times4.0833\approx60.0245 \)
Now, \( f(4.0833)=-30.0130 + 60.0245-27=3.0115\approx3.01 \)

Answer:

\( 3.01 \)