Let $g(x)=2 x^{3}-21 x^{2}+60 x$ . $\newline$ What is the absolute maximum value of $g$ over the closed interval $[0,6]$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $42$ $\newline$ (B) $25$ $\newline$ (C) $52$ $\newline$ (D) $36$

View step-by-step help

Home

Math Problems

Calculus

Euler's method

Full solution

Q. Let

g(x)=2 x^{3}-21 x^{2}+60 x

\newline

What is the absolute maximum value of

g

over the closed interval

[0,6]

\newline

Choose

1

answer:

\newline

(A)

42

\newline

(B)

25

\newline

(C)

52

\newline

(D)

36

Calculate Derivative of $g(x)$ : To find the absolute maximum value of the function $g(x) = 2x^3 - 21x^2 + 60x$ over the closed interval $[0,6]$ , we first need to find the critical points of $g(x)$ within the interval. Critical points occur where the derivative $g'(x)$ is zero or undefined. Let's calculate the derivative $g'(x)$ .
Find Critical Points: The derivative of $g(x)$ with respect to $x$ is $g'(x) = \frac{d}{dx} [2x^3 - 21x^2 + 60x]$ . Using the power rule, we find $g'(x) = 6x^2 - 42x + 60$ .
Solve Quadratic Equation: Next, we need to find the values of $x$ where $g'(x) = 0$ . Setting the derivative equal to zero gives us the equation $6x^2 - 42x + 60 = 0$ . We can simplify this by dividing the entire equation by $6$ , resulting in $x^2 - 7x + 10 = 0$ .
Evaluate Function at Points: We now solve the quadratic equation $x^2 - 7x + 10 = 0$ . This can be factored into $(x - 5)(x - 2) = 0$ , giving us two critical points: $x = 2$ and $x = 5$ .
Compare Values: We must evaluate $g(x)$ at the critical points and at the endpoints of the interval $[0,6]$ to find the absolute maximum. The endpoints are $x = 0$ and $x = 6$ . Let's calculate $g(0)$ , $g(2)$ , $g(5)$ , and $g(6)$ .
Compare Values: We must evaluate $g(x)$ at the critical points and at the endpoints of the interval $[0,6]$ to find the absolute maximum. The endpoints are $x = 0$ and $x = 6$ . Let's calculate $g(0)$ , $g(2)$ , $g(5)$ , and $g(6)$ .Evaluating $g(x)$ at $x = 0$ gives $[0,6]$ $0$ .
Compare Values: We must evaluate $g(x)$ at the critical points and at the endpoints of the interval $[0,6]$ to find the absolute maximum. The endpoints are $x = 0$ and $x = 6$ . Let's calculate $g(0)$ , $g(2)$ , $g(5)$ , and $g(6)$ .Evaluating $g(x)$ at $x = 0$ gives $[0,6]$ $0$ .Evaluating $g(x)$ at $[0,6]$ $2$ gives $[0,6]$ $3$ .
Compare Values: We must evaluate $g(x)$ at the critical points and at the endpoints of the interval $[0,6]$ to find the absolute maximum. The endpoints are $x = 0$ and $x = 6$ . Let's calculate $g(0)$ , $g(2)$ , $g(5)$ , and $g(6)$ .Evaluating $g(x)$ at $x = 0$ gives $[0,6]$ $0$ .Evaluating $g(x)$ at $[0,6]$ $2$ gives $[0,6]$ $3$ .Evaluating $g(x)$ at $[0,6]$ $5$ gives $[0,6]$ $6$ .
Compare Values: We must evaluate $g(x)$ at the critical points and at the endpoints of the interval $[0,6]$ to find the absolute maximum. The endpoints are $x = 0$ and $x = 6$ . Let's calculate $g(0)$ , $g(2)$ , $g(5)$ , and $g(6)$ .Evaluating $g(x)$ at $x = 0$ gives $[0,6]$ $0$ .Evaluating $g(x)$ at $[0,6]$ $2$ gives $[0,6]$ $3$ .Evaluating $g(x)$ at $[0,6]$ $5$ gives $[0,6]$ $6$ .Evaluating $g(x)$ at $x = 6$ gives $[0,6]$ $9$ .
Compare Values: We must evaluate $g(x)$ at the critical points and at the endpoints of the interval $[0,6]$ to find the absolute maximum. The endpoints are $x = 0$ and $x = 6$ . Let's calculate $g(0)$ , $g(2)$ , $g(5)$ , and $g(6)$ .Evaluating $g(x)$ at $x = 0$ gives $[0,6]$ $0$ .Evaluating $g(x)$ at $[0,6]$ $2$ gives $[0,6]$ $3$ .Evaluating $g(x)$ at $[0,6]$ $5$ gives $[0,6]$ $6$ .Evaluating $g(x)$ at $x = 6$ gives $[0,6]$ $9$ .Comparing the values of $g(x)$ at $x = 0$ , $[0,6]$ $2$ , $[0,6]$ $5$ , and $x = 6$ , we find that the largest value is $x = 0$ $5$ . Therefore, the absolute maximum value of $g(x)$ on the interval $[0,6]$ is $x = 0$ $8$ .