Chapter 6.8, Problem 31SGR

a.

To determine

To graph: The function $h\left(x\right)=4x^3-6x^2+1$ .

Explanation of Solution

Given information:

The polynomial $h\left(x\right)=4x^3-6x^2+1$ .

Graph:

Consider the equation $h\left(x\right)=4x^3-6x^2+1$

Construct a table of values,

  $\begin{array}{cc}x& g\left(x\right)\\ -3& -161\\ -2& -55\\ -1& -9\\ 0& 1\\ 1& -1\\ 2& 9\\ 3& 55\end{array}$

Connect the points obtained above to obtain a smooth curve.

  

Interpretation:

The function $h\left(x\right)=4x^3-6x^2+1$ has 3 real zeros at as the graph intersects the x -axis at three points.

b.

To determine

To calculate: The consecutive integer values of x at which real zero is located.

Answer to Problem 31SGR

The real zeros of the function are located between $x=-1\text{ and }x=0$, $x=0\text{ and }x=1$, $x=1\text{ and }x=2$ .

Explanation of Solution

Given information:

The function $h\left(x\right)=4x^3-6x^2+1$ .

Calculation:

Consider the function $h\left(x\right)=4x^3-6x^2+1$ .

Construct a table of values,

  $\begin{array}{cc}x& g\left(x\right)\\ -3& -161\\ -2& -55\\ -1& -9\\ 0& 1\\ 1& -1\\ 2& 9\\ 3& 55\end{array}$

Recall that if for a polynomial function and a and b are two real numbers such that $f\left(a\right)<0$ and $f\left(b\right)>0$ . Then the function has at least one real zero between a andb .

From the above table the changes in sign at integer values indicate that zeros of the function lie between $x=-1\text{ and }x=0$ , $x=0\text{ and }x=1$ , $x=1\text{ and }x=2$ .

c.

To determine

To calculate: The approximate x -coordinates where relative maxima and minima occur.

Answer to Problem 31SGR

The function has relative minima near $x=1$ and relative maxima near $x=0$ .

Explanation of Solution

Given information:

The function $h\left(x\right)=4x^3-6x^2+1$ .

Calculation:

Consider the function $h\left(x\right)=4x^3-6x^2+1$ .

Construct a table of values,

  $\begin{array}{cc}x& g\left(x\right)\\ -3& -161\\ -2& -55\\ -1& -9\\ 0& 1\\ 1& -1\\ 2& 9\\ 3& 55\end{array}$

The function has relative high and low values of function. A function has relative minimum when no other near by point has a lesser y -coordinate. A function has relative maximum when no other near by point has a greater y -coordinate.

Observe from the table that the polynomial function $h\left(x\right)=4x^3-6x^2+1$ has the value greater than the surrounding points at $x=0$ , so there must be a relative maximum at this point.

Similarly, the polynomial function $h\left(x\right)=4x^3-6x^2+1$ has the value less than the surrounding points at $x=1$ , so there must be a relative minimum at this point.

Thus, the function has relative minima near $x=1$ and relative maxima near $x=0$ .