Question
If \(p\) is a positive real constant, prove that \(y=e^{px^2}\) does not have any points of inflection.
Try the question yourself first. If you get stuck, open the hints before using the full walkthrough.
You must use calculus and show any derivatives that you need.
Step 1
Find the first derivative
Use the chain rule on \(y=e^{px^2}\).
Step 2
Find the second derivative
Differentiate \(2pxe^{px^2}\) using the product rule.
Step 3
Factorise the second derivative
Take out the common factors.
Step 4
Can the second derivative ever be zero?
Think about each factor in \(2pe^{px^2}(1+2px^2)\).
Step 5
State the conclusion
What does this tell us about points of inflection?
Step 6
Final proof and graph
A clean proof is:
\[
\frac{dy}{dx}=2pxe^{px^2}
\]
\[
\frac{d^2y}{dx^2}=2pe^{px^2}+4p^2x^2e^{px^2}
\]
\[
\frac{d^2y}{dx^2}=2pe^{px^2}(1+2px^2)
\]
\[
p>0,\quad e^{px^2}>0,\quad \text{and}\quad 1+2px^2>0
\]
\[
\therefore \frac{d^2y}{dx^2}>0 \text{ for all } x
\]
\[
\text{So } y=e^{px^2} \text{ has no points of inflection.}
\]