Question

find the maximum value of the function $f(x) = -1.3x^2 + 18.4x - 58$ 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, \( a=- 1.3 \), \( b = 18.4 \), \( c=-58 \).
So, \( x=-\frac{18.4}{2\times(-1.3)}=\frac{18.4}{2.6}\approx7.0769 \)

Step2: Substitute x into the function

Substitute \( x\approx7.0769 \) into \( f(x)=-1.3x^{2}+18.4x - 58 \)
\( f(7.0769)=-1.3\times(7.0769)^{2}+18.4\times7.0769-58 \)
First, calculate \( (7.0769)^{2}\approx50.07 \)
Then, \( - 1.3\times50.07\approx - 65.091 \)
\( 18.4\times7.0769\approx130.215 \)
Now, \( f(7.0769)=-65.091 + 130.215-58=7.124\approx7.12 \)

Answer:

\( 7.12 \)