How do you find the points of inflection of the curve y=e^{x^2}?

Question:

How do you find the points of inflection of the curve {eq}y=e^{x^2} {/eq}?

Infection Points:

An inflection point indicates the point of a function where it changes concavity. We can find the inflection points of a function when we set the second derivative of the function equal to zero. However, we must consider that the point must define a change in the concavity of the function; if the concavity of the function never changes, the second derivative will have no real roots.

Answer and Explanation:

We have the function

{eq}y=e^{x^2} \\ {/eq}

Taking the first derivative the function using the exponent and chain rules:

{eq}f'(x)=2\,x{{\rm e}^{{x}^{2}}} \\ {/eq}

{eq}f'(x)=0 {/eq} when {eq}x=0 {/eq}

Critical point at: {eq}(0, 1) \\ {/eq}

Now, taking the second derivative the function using the product rule:

{eq}f''(x)=2\,{{\rm e}^{{x}^{2}}} \left( 2\,{x}^{2}+1 \right) \\ {/eq}

Now setting it equal to zero {eq}f''(x)=0 {/eq}

we can see the function has no real roots:

{eq}2e^{x^2} \neq 0 {/eq} for all values of x

and

{eq}1+2x^2 = 0 \Rightarrow x=\sqrt{-\frac{1}{2}} {/eq}

Therefore, the function {eq}y=e^{x^2} {/eq} has no inflection points.


Learn more about this topic:

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Concavity and Inflection Points on Graphs

from Math 104: Calculus

Chapter 9 / Lesson 5
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