find ( f(-2) ) if ( f(x)=\frac{x^{3}}{7}-8x ).\n\n( f(-2)=square ) (simplify your answer. type an integer or…

find ( f(-2) ) if ( f(x)=\frac{x^{3}}{7}-8x ).\n\n( f(-2)=square ) (simplify your answer. type an integer or a fraction.)

find ( f(-2) ) if ( f(x)=\frac{x^{3}}{7}-8x ).\n\n( f(-2)=square ) (simplify your answer. type an integer or a fraction.)

Answer

Explanation:

Step1: Differentiate the function

Use the power rule ((x^n)^\prime = nx^{n - 1}). For (y=\frac{x^{3}}{7}-8x), the derivative (y^\prime=\frac{3x^{2}}{7}-8).

Step2: Substitute (x = - 2)

Substitute (x=-2) into (y^\prime). (y^\prime|_{x = - 2}=\frac{3\times(-2)^{2}}{7}-8). First calculate ((-2)^{2}=4), then (\frac{3\times4}{7}-8=\frac{12}{7}-8). Rewrite (8) as (\frac{56}{7}), so (\frac{12}{7}-\frac{56}{7}=\frac{12 - 56}{7}). (12-56=-44), then (\frac{-44}{7}).

Answer:

(-\frac{44}{7})