find the critical points of f. assume a is a constant. f(x)=1/21x^21 - a^20x select the correct choice below…

find the critical points of f. assume a is a constant. f(x)=1/21x^21 - a^20x select the correct choice below and fill in any answer boxes within your choice. a. x= (use a comma to separate answers as needed.) b. f has no critical points.

find the critical points of f. assume a is a constant. f(x)=1/21x^21 - a^20x select the correct choice below and fill in any answer boxes within your choice. a. x= (use a comma to separate answers as needed.) b. f has no critical points.

Answer

Explanation:

Step1: Find the derivative of f(x)

Using the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, if $f(x)=\frac{1}{21}x^{21}-a^{20}x$, then $f^\prime(x)=\frac{1}{21}\times21x^{20}-a^{20}=x^{20}-a^{20}$.

Step2: Set the derivative equal to zero

Set $f^\prime(x) = 0$, so $x^{20}-a^{20}=0$. We can rewrite it as $(x^{10}+a^{10})(x^{10}-a^{10}) = 0$ using the difference - of - squares formula $A^2 - B^2=(A + B)(A - B)$. Then, $x^{10}+a^{10}=0$ gives $x^{10}=-a^{10}$ (no real solutions when $a\in R$) and $x^{10}-a^{10}=0$ gives $x^{10}=a^{10}$, so $x=\pm a$.

Answer:

A. $x = a,-a$