Question

find the absolute maximum and absolute minimum of (f(x) = x^3 - 3x^2 - 72x - 29) over the interval (-3, 7)

the absolute maximum over the interval (-3, 7) will be \\(\square\\) , which will occur at (x = \square)

the absolute minimum over the interval (-3, 7) will be \\(\square\\) , which will occur at (x = \square)

Explanation:

Find the derivative of the function

Using the Critical Points knowledge point

$$ f'(x) = 3x^2 - 6x - 72 $$

Find the critical points in the interval

Using the Critical Points knowledge point

$$
LATEXBLOCK0
$$

Evaluate the function at the critical point and endpoints

Using the Absolute Extrema on Closed Intervals knowledge point

$$
LATEXBLOCK1
$$

Determine the absolute maximum and minimum

Using the Absolute Extrema on Closed Intervals knowledge point

$$
LATEXBLOCK2
$$

Answer:

Find the absolute maximum and absolute minimum of \(f(x) = x^3 - 3x^2 - 72x - 29\) over the interval \([-3,7]\)

The absolute maximum over the interval \([-3, 7]\) will be <blank>133</blank>, which will occur at \(x =\) <blank>-3</blank>

The absolute minimum over the interval \([-3, 7]\) will be <blank>-353</blank>, which will occur at \(x =\) <blank>6</blank>