Find the absolute maximum of the function $f ( x ) = x ^ { 3 } - x ^ { 2 } - x$ on the interval $- 10 \leq x \leq 2$ .
A) $\frac { 1 } { 9 }$

B) $\frac { 1 } { 5 }$
C)0
D) $\frac { 1 } { 7 }$
E)1
F) $\frac { 1 } { 4 }$
G) $\frac { 1 } { 3 }$

H)2

Question

Find the absolute maximum of the function $f ( x ) = x ^ { 3 } - x ^ { 2 } - x$ on the interval $- 10 \leq x \leq 2$ .
A) $\frac { 1 } { 9 }$

B) $\frac { 1 } { 5 }$ 
C)0
D) $\frac { 1 } { 7 }$ 
E)1
F) $\frac { 1 } { 4 }$ 
G) $\frac { 1 } { 3 }$

H)2

Quizplus · Accepted Answer

To find the absolute maximum of the function $f(x) = x^3 - x^2 - x$ on the interval $[-10, 2]$, we first find its critical points by taking the derivative and setting it equal to zero. The derivative is $f'(x) = 3x^2 - 2x - 1$. Solving $3x^2 - 2x - 1 = 0$ does not give any critical points that fall within the interval $[-10, 2]$. Therefore, the absolute maximum must occur at one of the endpoints of the interval. Evaluating the function at these endpoints, we get $f(-10) = -10^3 - (-10)^2 - (-10) = -1000 + 100 + 10 = -890$ and $f(2) = 2^3 - 2^2 - 2 = 8 - 4 - 2 = 2$. Since $2 > -890$, the absolute maximum value of the function on the interval $[-10, 2]$ is $2$, which corresponds to choice H.

Find the Absolute Maximum of the Function f(x)=x3−x2−xf ( x ) = x ^ { 3 } - x ^ { 2 } - xf(x)=x3−x2−x

Find the Absolute Maximum of the Function $f ( x ) = x ^ { 3 } - x ^ { 2 } - x$