Science:Math Exam Resources/Courses/MATH101/April 2014/Question 01 (j)

MATH101 April 2014

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Question 01 (j)
Short Problem. Show your work. No credit will be given for the answer without the correct accompanying work. Let $f(x)=\int _{1}^{x}100(t^{2}-3t+2)e^{-t^{2}}\,dt$. Find the interval(s) on which ƒ is increasing.

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
Recall that a function is increasing when its derivative is positive.

Hint 2
Consider the Fundamental Theorem of Calculus

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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. Note that $f$ is increasing on some interval $(a,b)$ if and only if $f'(x)>0$ on $(a,b)$. By the Fundamental theorem of Calculus, we have ${\begin{aligned}f'(x)=100(x^{2}-3x+2)e^{-x^{2}}.\end{aligned}}$ Since $e^{-x^{2}}$ is always positive, we only need to check when the quadratic polynomial $x^{2}-3x+2$ is positive. We can factor this to get $x^{2}-3x+2=(x-2)(x-1).$ This will be positive when both factors are positive or both factors are negative. Both factors are positive when $x>2$ and both factors are negative when $x<1$. Therefore the function is increasing when $x<1$ or $x>2$.

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