Question

the function (f(x) = 2x^3 - 45x^2 + 300x + 10) has one local minimum and one local maximum. use a graph of the function to estimate these local extrema.

this function has a local minimum at (x =) with output value

and a local maximum at (x =) with output value

Explanation:

Find the first derivative of the function

$$
LATEXBLOCK0
$$

Find the critical points by setting the derivative to zero

$$
LATEXBLOCK1
$$

Evaluate the function at the critical points to classify the local extrema

$$
LATEXBLOCK2
$$

Since \(f(5) > f(10)\), the local maximum is at \(x = 5\) with value \(635\), and the local minimum is at \(x = 10\) with value \(510\).

Answer:

The function \(f(x) = 2x^3 - 45x^2 + 300x + 10\) has one local minimum and one local maximum. Use a graph of the function to estimate these local extrema.

This function has a local minimum at \(x =\) <blank>10</blank> with output value <blank>510</blank>

and a local maximum at \(x =\) <blank>5</blank> with output value <blank>635</blank>