Science:Math Exam Resources/Courses/MATH104/December 2011/Question 01 (j)

MATH104 December 2011

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q1 (d)  •  Q1 (e)  •  Q1 (f)  •  Q1 (g)  •  Q1 (h)  •  Q1 (i)  •  Q1 (j)  •  Q1 (k)  •  Q1 (l)  •  Q1 (m)  •  Q1 (n)  •  Q2 (a)  •  Q2 (b)  •  Q2 (c)  •  Q2 (d)  •  Q2 (e)  •  Q2 (f)  •  Q2 (g)  •  Q3  •  Q4  •  Q5 (a)  •  Q5 (b)  •  Q5 (c)  •  Q5 (d)  •  Q6 (a)  •  Q6 (b)  •  Q6 (c)  •  Q6 (d)  •  Q6 (e)  •  

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Question 01 (j)
Let $f$ and $g$ be functions such that $\displaystyle f'(x)$ and $\displaystyle g'(x)$ are both continuous and increasing for all real numbers $x$. Which of the following is ALWAYS true? $f+g$ is concave up everywhere $f-g$ is concave up everywhere $f\cdot g$ is concave up everywhere I only II only I and II only I and III only I, II, and III

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
What do you know about $f''(x)$ and $g''(x)$ from the statement of the problem?

Hint 2
To test (I), define a function $H=f+g$. What is the value of the second derivative of $H$? You can use a similar method to test II and III.

Checking a solution serves two purposes: helping you if, after having used all the hints, 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. We can consider statements I, II, III individually to which of them are true. Statement I: f + g is Concave Up Everywhere Let $\displaystyle H=f+g$ Statement I is true if the second derivative of $H$ is positive for all real numbers $x$. The second derivative of $H$ is $\displaystyle H''=f''+g''.$ However, $\displaystyle f''(x)$ and $\displaystyle g''(x)$ must be positive for all real numbers (because $f'$ and $g'$ are continuous and increasing for all $x$), so $\displaystyle H''(x)$ must be positive for all $x$. Statement I is true. Statement II: f - g is Concave Up Everywhere If we let $\displaystyle H=f-g$ then $\displaystyle H''=f''-g''.$ However, $\displaystyle f''(x)$ could be less than $\displaystyle g''(x)$ for all real numbers, so $\displaystyle H''(x)$ may not be positive for all $x$. Statement II is false. Statement III: fg is Concave Up Everywhere If we let $\displaystyle H=f\cdot g$ then $\displaystyle H''=f''g+2f'g'+fg''.$ We know that $\displaystyle f''(x),g''(x)$ are both positive for all real numbers, but we don't necessarily have that $\displaystyle f'(x),g'(x),f(x),g(x)$ are necessarily positive. So it is possible that $f''g+fg''$ is negative and outweighs the positive term $2f'g'$. For example, choose $\displaystyle f(x)=x^{2}-1,\ g(x)=x^{2}$ Then $\displaystyle f'(x)=g'(x)=2x$ is increasing but $\displaystyle f''g+2f'g'+2fg''=(2)(x^{2})+2(2x)(2x)+(x^{2}-1)(2)=12x^{2}-2$ which is negative for small values of $x$ (e.g. when $x=0$). Statement III is false. The Answer The answer is (A) I only.

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