Science:Math Exam Resources/Courses/MATH100/December 2011/Question 05 (b)

MATH100 December 2011

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Question 05 (b)
Full-Solution Problems. In questions 2-8, justify your answers and show all your work. Simplification of answers is not required unless explicitly stated. Let $\displaystyle y=f(x)=6x^{1/5}+x^{6/5}.$ Find the interval(s) of decrease.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Derivatives can help us detect where functions are decreasing. Try determining when the derivative is negative.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The intervals of decrease are the intervals on which the derivative of the function ƒ is negative. Given that ${\begin{aligned}f'(x)&={\frac {6}{5}}x^{-4/5}+{\frac {6}{5}}x^{1/5}\\&={\frac {6x^{-4/5}(1+x)}{5}}\\&={\frac {6(x+1)}{5x^{4/5}}}\end{aligned}}$ Our critical points occur when $x=-1,0$, that is, points on the function where the derivative is undefined or zero. Writing out a sign chart, we see that when $x<-1$ we have that $f'(x)<0$. When $-1<x<0$ we have that $f'(x)>0$. When $0<x$ we have that $f'(x)>0$. Hence the function is decreasing on $(-\infty ,-1)$.

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