What are the critical points of $g(x)=x^(2/5)(x-10)$ ?

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Answer 1 · 2015-10-29T03:34:06

$0$ and $20/7$

$g(x)=x^(2/5)(x-10)$

Note first that $"Dom(g) = (-oo,oo)$ so every point at which $f'(x)$ is undefined or $0$ is a critical number.

Let's find $g'(x)$ using the product rule:

$g'(x) = 2/5 x^(-3/5)(x-10) + x^(2/5)(1)$ .

Let's now factor the denominators out of the expression:

$g'(x) = 1/5 x^(-3/5)[2(x-10)+5x]$ .

We'll simplify the brackets and write the denominator as a denominator:

$g'(x) = (7x-20)/(5 x^(3/5)$ .

$g'(x)$ is not defined for $x=0$ and is $0$ at $x=20/7$ .

Both are in $"Dom"(g)$ , so both are critical numbers for $g$ .

What are the critical points of g(x)=x^(2/5)(x-10)g(x)=x^(2/5)(x-10)?