Find the absolute maximum of the function $f ( x ) = x ^ { 7 } - x ^ { 6 }$ on the interval -1$\le$x$\le$1.
A) -2
B) 0
C) 1
D) -1

Question

Quizplus · Accepted Answer

The function $f(x) = x^7 - x^6$ has a derivative $f'(x) = 7x^6 - 6x^5$. Setting $f'(x) = 0$ gives $x(7x^5 - 6x^4) = 0$, which has solutions at $x = 0$ and $x = \frac{6}{7}$. Evaluating $f(x)$ at the critical points and the endpoints of the interval $[-1, 1]$, we find $f(-1) = -2$, $f(0) = 0$, $f(\frac{6}{7})$ is less than 1, and $f(1) = 0$. Therefore, the absolute maximum of $f(x)$ on the interval $[-1, 1]$ is 0.

Find the Absolute Maximum of the Function f(x)=x7−x6f ( x ) = x ^ { 7 } - x ^ { 6 }f(x)=x7−x6

Find the Absolute Maximum of the Function $f ( x ) = x ^ { 7 } - x ^ { 6 }$