If `a+2b+3c=4,` then find the least value of `a^2+b^2+c^2dot`

Explanation

To solve the problem of finding the least value of \( a^2 + b^2 + c^2 \) given the constraint \( a + 2b + 3c = 4 \), we can use the method of Lagrange multipliers or the Cauchy-Schwarz inequality. Here, we will use the Cauchy-Schwarz inequality for simplicity.

### Step-by-Step Solution:

1. **Understanding the Constraint**: We have the equation \( a + 2b + 3c = 4 \). This can be thought of as a linear constraint on the variables \( a, b, \) and \( c \).

2. **Applying Cauchy-Schwarz Inequality**: According to the Cauchy-Schwarz inequality:
\[
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