Q.Find the intervals of convexity and concavity of the Gaussian curve $y=e^{-x^2}$ and also find the points of inflection. OR Show that $(Z, *)$ is an infinite abelian group, where '*' is defined as $a*b=a+b+2$ and Z is the set of all integers.

Main part: use the sign of $y''$ for $y=e^{-x^2}$ to find convex/concave intervals and inflection points. OR alternative: verify the group axioms for $(Z,*)$ with $a*b=a+b+2$.

Main part — convexity, concavity and inflection points of $y=e^{-x^2}$

1. First derivative.

   $y=e^{-x^2}\Rightarrow y'=-2x\,e^{-x^2}$

2. Second derivative (product rule on $-2x$ and $e^{-x^2}$):

   $y''=-2e^{-x^2}+(-2x)\big(-2x\,e^{-x^2}\big)=-2e^{-x^2}+4x^2e^{-x^2}=e^{-x^2}(4x^2-2)=2e^{-x^2}(2x^2-1)$

3. Find where $y''=0$. Since $e^{-x^2}>0$ for all $x$, $y''=0 \iff 2x^2-1=0 \iff x^2=\dfrac12 \iff x=\pm\dfrac{1}{\sqrt2}$.

4. Determine the sign of $y''$ on each interval (sign follows the sign of $2x^2-1$, since $e^{-x^2}>0$ always):

   • For $|x|>\dfrac{1}{\sqrt2}$: $2x^2-1>0 \Rightarrow y''>0$ — curve is convex (concave up).

   • For $|x|<\dfrac{1}{\sqrt2}$: $2x^2-1<0 \Rightarrow y''<0$ — curve is concave (concave down).

5. Points of inflection. At $x=\pm\dfrac{1}{\sqrt2}$, $y''$ changes sign, so these are genuine inflection points.

   $y\left(\pm\dfrac{1}{\sqrt2}\right)=e^{-1/2}=\dfrac{1}{\sqrt e}$

   Inflection points: $\left(\dfrac{1}{\sqrt2},\dfrac{1}{\sqrt e}\right)$ and $\left(-\dfrac{1}{\sqrt2},\dfrac{1}{\sqrt e}\right)$.

OR — alternative: $(Z,*)$ with $a*b=a+b+2$ is an infinite abelian group

1. Closure. For any $a,b\in Z$, $a*b=a+b+2$ is a sum of integers plus $2$, hence an integer. So $a*b\in Z$ — closure holds.

2. Associativity. For any $a,b,c\in Z$:

   $(a*b)*c=(a+b+2)*c=(a+b+2)+c+2=a+b+c+4$

   $a*(b*c)=a*(b+c+2)=a+(b+c+2)+2=a+b+c+4$

   Both equal $a+b+c+4$, so $(a*b)*c=a*(b*c)$ — associativity holds.

3. Identity element. Seek $e\in Z$ with $a*e=a$ for all $a$: $a+e+2=a\Rightarrow e=-2$.

   Check: $e*a=-2+a+2=a$, and $a*e=a+(-2)+2=a$. So $e=-2\in Z$ is a two-sided identity.

4. Inverse of each element. For $a\in Z$, seek $a'\in Z$ with $a*a'=e=-2$: $a+a'+2=-2\Rightarrow a'=-4-a=-(a+4)$.

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