Question

given the function ( f(x) = x^{5}+20x^{4} ), determine all intervals on which ( f ) is decreasing

Explanation:

Step1: Find critical points

Set \(f'(x)=x^{5}+20x^{4}=x^{4}(x + 20)=0\).
Solving \(x^{4}(x + 20)=0\), we get \(x = 0\) or \(x=-20\).

Step2: Test intervals

• For \(x<-20\), let \(x=-21\). Then \(f'(-21)=(-21)^{4}(-21 + 20)=(-21)^{4}\times(-1)<0\).
• For \(-20
• For \(x>0\), let \(x = 1\). Then \(f'(1)=1^{4}(1 + 20)=21>0\).

Answer:

The function \(f(x)\) is decreasing on the interval \((-\infty,-20)\).