Find the greatest value of `x^2y^3z^4` if `x^2+y^2+z^2=1,w h e r ex ,y ,z` are positive.

Explanation

To find the greatest value of \( x^2 y^3 z^4 \) given the constraint \( x^2 + y^2 + z^2 = 1 \) where \( x, y, z \) are positive, we can apply the method of Lagrange multipliers or use the AM-GM inequality. Here, we will use the AM-GM inequality.

### Step-by-Step Solution:

1. **Apply the AM-GM Inequality**:
We can express \( x^2, y^2, z^2 \) in a way that allows us to use the AM-GM inequality. We rewrite \( x^2 \) as \( \frac{x^2}{2} + \frac{x^2}{2} \), \( y^2 \) as \( \frac{y^2}{3} + \frac{y^2}{3} + \frac{y^2}{3} \), and \( z^2 \) as \( \frac{z^2}{4} + \frac{z^2}{4} + \frac{z^2}{4} + \frac{z^2}{4} \).

This gives us:
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