Chapter SH, Problem 9.1.15EP

(a)

To determine

Whether the function has maximum or minimum value.

Answer to Problem 9.1.15EP

The function has a minimum value.

Explanation of Solution

Given:

The given function is:

  $y=6x^2-12x-4$

Calculation:

Finding the first and second derivatives of the given function:

  $\begin{array}{l}y=6x^2-12x-4\\ \frac{dy}{dx}=12x-12\\ \frac{d^{2}y}{d^{2}x}=12\\ Also,\\ 0=12x-12\\ 12x=12\\ x=1\end{array}$

Calculating the value of $y=6x^2-12x-4$ .

  $\begin{array}{l}y=6{\left(1\right)}^2-12\left(1\right)-4\\ y=6-12-4\\ y=-10\end{array}$

The second derivative of the given function is positive. Hence, the function has a minimum value.

(b)

To determine

The maximum or minimum value.

Answer to Problem 9.1.15EP

  $\left(1,-10\right)$

Explanation of Solution

Given:

The given function is:

  $y=6x^2-12x-4$

Calculation:

The minimum value is:

  $\left(1,-10\right)$

(c)

To determine

The domain and range of the given function.

Answer to Problem 9.1.15EP

The domain will be the set of all real numbers and range will be $\{y:y\ge -10\}.$

Explanation of Solution

Given:

The given function is:

  $y=6x^2-12x-4$

Calculation:

The graph of the given function will be:

  

The domain will be the set of all real numbers and range will be $\{y:y\ge -10\}.$