Science:Math Exam Resources/Courses/MATH100/December 2016/Question 07 (b)

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MATH100 December 2016

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Question 07 (b)
Which of the following five functions are concave up on their whole domain? Select all that apply. $f(x)=\cos(x)+x^{2}$ $f(x)=e^{-x^{2}}$ $f(x)=x^{4}+2x^{2}-5x+1$ $f(x)=1-x^{2}$ $f(x)=-\log(x)$ . Recall $\log x=\log _{e}x=\ln x$ .

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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
A twice-differentiable function $f$ is concave up (i.e., convex) on an interval $I$ if and only if $f''(x)\geq 0$ for all $x\in I$ .

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Solution
Differentiate each function twice. We then get the following: $f''(x)=2-\cos(x)$ — always nonnegative, because $-1\leq \cos(x)\leq 1$ $f'(x)=-2xe^{-x^{2}}\Rightarrow f''(x)=4x^{2}e^{-x^{2}}-2e^{-x^{2}}=2(2x^{2}-1)e^{-x^{2}}$ — changes sign $f'(x)=4x^{3}+4x-5\Rightarrow f''(x)=12x^{2}+4$ — always nonnegative $f''(x)=-2$ — always negative $f'(x)=-1/x\Rightarrow f''(x)=1/x^{2}$ — always nonnegative. Therefore, by the hint, the answers are: $\color {blue}{\text{(A), (C), (E)}}$