Chapter SH, Problem 9.1.14EP

(a)

To determine

Whether the function has maximum or minimum value.

Answer to Problem 9.1.14EP

The function has a maximum value.

Explanation of Solution

Given:

The given function is:

  $y=-2x^2-2x+4$

Calculation:

Finding the first and second derivatives of the given function:

  $\begin{array}{l}y=-2x^2-2x+4\\ \frac{dy}{dx}=-4x-2\\ \frac{d^{2}y}{d^{2}x}=-4\\ Also,\\ 0=-4x-2\\ 4x=-2\\ x=-\frac{1}{2}\end{array}$

Calculating the value of $y=-2x^2-2x+4$ .

  $\begin{array}{l}y=-2{\left(-\frac{1}{2}\right)}^2-2\left(-\frac{1}{2}\right)+4\\ y=-\frac{1}{2}+1+4\\ y=\frac{9}{2}\end{array}$

The second derivative of the given function is negative. Hence, the function has a maximum value.

(b)

To determine

The maximum or minimum value.

Answer to Problem 9.1.14EP

  $\left(-\frac{1}{2},\frac{9}{2}\right)$

Explanation of Solution

Given:

The given function is:

  $y=-2x^2-2x+4$

Calculation:

The maximum value is:

  $\left(-\frac{1}{2},\frac{9}{2}\right)$

(c)

To determine

The domain and range of the given function.

Answer to Problem 9.1.14EP

The domain will be the set of all real numbers and range will be $\{y:y\le \frac{9}{2}\}.$

Explanation of Solution

Given:

The given function is:

  $y=-2x^2-2x+4$

Calculation:

The graph of the given function will be:

  

The domain will be the set of all real numbers and range will be $\{y:y\le \frac{9}{2}\}.$