How to Find the X-Coordinate of the Stationary Point on the Curve y 2x^5e^{-2x}

Understanding how to find the stationary points of a curve is an essential skill in calculus and differential calculus. In this article, we'll walk through the process of finding the x-coordinate of the stationary point for the curve defined by the function y 2x^5e^{-2x}.

Step-by-Step Guide to Finding Stationary Points

Stationary points occur where the derivative of a function equals zero. This is crucial for identifying local maxima, minima, or points of inflection.

Step 1: Differentiating the Function

To find the stationary points, we first need to differentiate the function. The function given is:

$$ y 2x^5e^{-2x} $$

We will use the product rule for differentiation. Recall that the product rule states:

$$ (uv)' uv' vu' $$

Let us define:

$$ u 2x^5 $$ $$ v e^{-2x} $$

Now, let's find the derivatives of u and v:

$$ u' 1^4 $$ $$ v' -2e^{-2x} $$

Applying the product rule:

$$ y' (2x^5)(-2e^{-2x}) (1^4)(e^{-2x}) $$

Simplifying this expression:

$$ y' e^{-2x}(1^4 - 4x^5) $$

Step 2: Setting the Derivative Equal to Zero

To find the stationary points, we set the derivative equal to zero:

$$ e^{-2x}(1^4 - 4x^5) 0 $$

Since e^{-2x} is never zero, we focus on:

$$ 1^4 - 4x^5 0 $$

Factoring out x^4:

$$ x^4(10 - 4x) 0 $$

This equation is zero when:

$$ x^4 0 $$ $$ 10 - 4x 0 $$

From x^4 0 we get:

$$ x 0 $$

From 10 - 4x 0 we get:

$$ x 2.5 $$

Conclusion

The x-coordinates of the stationary points on the curve y 2x^5e^{-2x} are:

$$ x 0 $$ $$ x 2.5 $$

Advanced Techniques and Calculations

For certain functions, solving for x exactly can be challenging. In such cases, numerical methods or software tools (like Wolfram Alpha) can be used to find the x-coordinate of the stationary points. For the function y 2x^5e^{-2x}, using numerical methods, we find that:

x ≈ 0.3816

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