Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 14

MATH100 A December 2023

  •  Q1  •  Q2  •  Q3  •  Q4  •  Q5  •  Q6  •  Q7  •  Q8  •  Q9  •  Q10  •  Q11  •  Q12  •  Q13  •  Q14  •  Q15  •  Q16  •  Q17  •  Q18  •  Q19  •  Q20  •  Q21  •  Q22  •  Q23  •  Q24  •  Q25  •  Q26  •  Q27(a)  •  Q27(b)  •  Q27(c)  •  Q28(a)  •  Q28(b)  •  Q29(a)  •  Q29(b)  •  Q30  •  

   Other MATH100 A Exams

  •  December 2023  •  

Question 14
Let ${\textstyle f(x)=x^{4}-4x^{3}-18x^{2}+13}$. Find all intervals where ${\textstyle f(x)}$ is concave up.

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
Recall that the second derivative tells you information about the concavity of a function.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

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. Recall how the second derivative gives us information about the concavity of a function: a function $f$ is concave up wherever the second derivative $f''$ is positive and concave down wherever the second derivative is negative. Thus the concavity changes at the points where the second derivative vanishes. First, let’s compute some derivatives: $${\begin{aligned}f'(x)&=4x^{3}-12x^{2}-36x,\\f''(x)&=12x^{2}-24x-36.\end{aligned}}$$ Now, set the second derivative to zero and solve for the points where the concavity changes: $${\begin{aligned}f''(x)=12x^{2}-24x-36&=0\\12(x^{2}-2x-3)&=0\\(x-3)(x+1)&=0.\end{aligned}}$$ So, we now have three intervals: ${\textstyle (-\infty ,-1)}$, ${\textstyle (-1,3)}$, and ${\textstyle (3,\infty )}$, and we need to determine in which ones the function is concave up. We can do this by evaluating the second derivative at a point in the interval to see if it’s positive or negative. ${\textstyle f''(-2)=60>0}$, ${\textstyle f''(0)=-36<0}$, and ${\textstyle f''(4)=60>0}$, so we can conclude that ${\textstyle f(x)}$ is concave up over the intervals ${\textstyle (-\infty ,-1)}$ and ${\textstyle (3,\infty )}$.