Question

use estimation use a graphing calculator to estimate the x-coordinates at which any extrema occur for the given function. round to the nearest hundredth. if there is no relative maximum or no relative minimum, write none. ( f(x) = x^3 + 3x^2 - 6x - 6 ) relative maximum ( x = ) blank relative minimum ( x = ) blank

Explanation:

Step1: Rewrite the function

The function is \( f(x)=x^{2}+3x^{2}-6x - 8 \), first combine like terms: \( f(x)=4x^{2}-6x - 8 \). This is a quadratic function in the form \( f(x)=ax^{2}+bx + c \), where \( a = 4 \), \( b=-6 \), \( c = -8 \).

Step2: Find the vertex (for minimum)

For a quadratic function \( y = ax^{2}+bx + c \), the x - coordinate of the vertex (which is the minimum point since \( a=4>0 \)) is given by \( x=-\frac{b}{2a} \).
Substitute \( a = 4 \) and \( b=-6 \) into the formula: \( x=-\frac{-6}{2\times4}=\frac{6}{8}=0.75 \).
To find the y - coordinate (the minimum value), substitute \( x = 0.75 \) into \( f(x) \):
\( f(0.75)=4\times(0.75)^{2}-6\times(0.75)-8 \)
\(=4\times0.5625-4.5 - 8 \)
\(=2.25-4.5 - 8=-10.25 \)
Since the coefficient of \( x^{2} \) is positive, the parabola opens upwards, so there is a relative minimum at \( x = 0.75 \) and no relative maximum.

Answer:

Relative minimum \( x = 0.75 \), Relative maximum: None (or no relative maximum)