Chapter 9.1, Problem 15CYU

a.

To determine

To decipher whether the function has a maximum or minimum value.

Answer to Problem 15CYU

The given function has a maximum value.

Explanation of Solution

Given information :

The function provided is $y=-3x^2+6x+3$ .

For the function $y=-3x^2+6x+3$ , the coefficients are $a=-3$ , $b=6$ and $c=3$ .

Since the coefficient of ‘a’ is negative, the curve of this function is downward facing and hence the function has a maximum value.

b.

To determine

To determine the maximum or minimum value of the given function.

Answer to Problem 15CYU

The function is maximum at 6 .

Explanation of Solution

Given information :

The function provided is $y=-3x^2+6x+3$ .

Formula used :

Formula to compute the x-coordinate of the vertex is $x=-\frac{b}{2\times a}$ . Here, $a=-3$ and $b=6$ . ‘a’ and ‘b’ are coefficients $x^2$ and $x$ respectively. The corresponding value of ‘y’ for the vertex would be the maximum or minimum.

Calculation :

Since the coefficient of ‘a’ is negative, the curve of this function is downward facing and hence the function has a maximum value.

X-coordinate of the vertex is

  $x=-\frac{b}{2\times a}$

Formula for axis of symmetry.

  $x=-\frac{6}{2\times \left(-3\right)}$

Putting the values of ‘a’ and ‘b’ .

  $=\frac{-6}{-6}$

Simplifying

  $x=1$

Vertex can be found out by putting the value of x computed in the axis of symmetry in the original function. This will give a value of y , that will be the maximum.

  $=-3{\left(1\right)}^2+6\left(1\right)+3$

Putting the value of $x=1$ in the original function.

  $=-3+6+3$

Simplifying the expression.

  $=6$

Thus, the maximum is at 6 .

c.

To determine

To calculate the domain and range for the given function.

Answer to Problem 15CYU

The domain is all real numbers, that is, $\left\{-\infty <x<\infty \right\}$ and range is all values lesser than or equal to 6 , that is, $\left\{y|y\le 6\right\}$ .

Explanation of Solution

Given information :

The function provided is $y=-3x^2+6x+3$ .

Formula used :

Formula to compute the x-coordinate of the vertex is $x=-\frac{b}{2\times a}$ . Here, $a=-3$ and $b=6$ . ‘a’ and ‘b’ are coefficients $x^2$ and $x$ respectively. The corresponding value of ‘y’ for the vertex would be the maximum or minimum.

Calculation :

Since the coefficient of ‘a’ is negative, the curve of this function is downward facing and hence the function has a maximum value.

X-coordinate of the vertex is

  $x=-\frac{b}{2\times a}$

Formula for axis of symmetry.

  $x=-\frac{6}{2\times \left(-3\right)}$

Putting the values of ‘a’ and ‘b’ .

  $=\frac{-6}{-6}$

Simplifying

  $x=1$

Vertex can be found out by putting the value of x computed in the axis of symmetry in the original function. This will give a value of y , that will be the maximum.

  $=-3{\left(1\right)}^2+6\left(1\right)+3$

Putting the value of $x=1$ in the original function.

  $=-3+6+3$

Simplifying the expression.

  $=6$

Thus, the maximum is at 6 .

Since x can take up any real values, the range is $\left\{-\infty <x<\infty \right\}$ .

Since the domain is the values of y, it can only take up values lesser than or equal to the maximum. That is, $\left\{y|y\le 6\right\}$ .