Question

the function \\(f(x) = -2x^3 + 45x^2 - 300x + 9\\) has one local minimum and one local maximum.
this function has a local minimum at \\(x =\\)
with value
and a local maximum at \\(x =\\)
with value

Explanation:

Find the derivative of the function

Using the Critical Points knowledge point

$$ f'(x) = -6x^2 + 90x - 300 $$

Find the critical points

Using the Critical Points knowledge point

$$
LATEXBLOCK0
$$

Classify the local extrema

Using the Local Extrema knowledge point

$$
LATEXBLOCK1
$$

Calculate the local minimum value

Using the Local Extrema knowledge point

$$
LATEXBLOCK2
$$

Calculate the local maximum value

Using the Local Extrema knowledge point

$$
LATEXBLOCK3
$$

Answer:

The function \(f(x) = -2x^3 + 45x^2 - 300x + 9\) has one local minimum and one local maximum.
This function has a local minimum at \(x =\) <blank>5</blank> with value <blank>-616</blank> and a local maximum at \(x =\) <blank>10</blank> with value <blank>-491</blank>.