Questions: Find the absolute maximum and minimum values of the function over the indicated interval, and indicate the (x)-values at which they occur. [ f(x)=2-6 x-6 x^2 ;[-3,3] ] The absolute maximum value is at (x=) (Use a comma to separate answers as needed.)

Find the absolute maximum and minimum values of the function over the indicated interval, and indicate the (x)-values at which they occur.
[ f(x)=2-6 x-6 x^2 ;[-3,3] ]

The absolute maximum value is at (x=)
(Use a comma to separate answers as needed.)

Transcript text: Find the absolute maximum and minimum values of the function over the indicated interval, and indicate the $x$-values at which they occur. \[ f(x)=2-6 x-6 x^{2} ;[-3,3] \] The absolute maximum value is $\square$ at $x=$ $\square$ (Use a comma to separate answers as needed.)

Solution

Solution Steps

Step 1: Find the derivative of the function

The derivative of the function, \(f'(x)\), is: \( - 12 x - 6 \)

Step 2: Solve \(f'(x) = 0\) for critical points

The critical points are found at: \( \left\{- \frac{1}{2}\right\} \)

Step 3 & 4: Evaluate \(f(x)\) at critical points and endpoints

At \(x = - \frac{1}{2}\), \(f(x) = 3.5\) At the endpoints, \(f(a) = -34\) and \(f(b) = -70\)

Step 5: Compare all evaluated values to find absolute maximum and minimum

The absolute maximum value is \( 3.5 \) at \(x = - \frac{1}{2}\) The absolute minimum value is \( -70 \) at \(x = 3\)

Final Answer:

The absolute maximum value of the function over the interval is \( 3.5 \) at \(x = - \frac{1}{2}\), and the absolute minimum value is \( -70 \) at \(x = 3\).

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