Find the greatest and the least values of the following functions on the indicated intervals ,
`y = x+ sqrt(x) " on " [0,4] `

Answer

To find the greatest and least values of the function \( y = x + \sqrt{x} \) on the interval \([0, 4]\), we will follow these steps:

### Step 1: Differentiate the function
We start by differentiating the function to determine its behavior (increasing or decreasing).

\[ y = x + \sqrt{x} \]

The derivative \( \frac{dy}{dx} \) is calculated as follows:

\[ \frac{dy}{dx} = \frac{d}{dx}(x) + \frac{d}{dx}(\sqrt{x}) = 1 + \frac{1}{2\sqrt{x}} \]

### Step 2: Analyze the derivative
Next, we analyze the derivative to determine where the function is increasing or decreasing.

The term \( \frac{1}{2\sqrt{x}} \) is always non-negative for \( x \geq 0 \). Therefore, \( \frac{dy}{dx} \) is always greater than or equal to 1 for \( x \in [0, 4] \):

\[ \frac{dy}{dx} = 1 + \frac{1}{2\sqrt{x}} \geq 1 \]

Since the derivative is always positive in the interval \([0, 4]\), the function \( y = x + \sqrt{x} \) is strictly increasing on this interval.

### Step 3: Evaluate the function at the endpoints
To find the least and greatest values of the function on the interval \([0, 4]\), we evaluate the function at the endpoints of the interval.

1. Evaluate at \( x = 0 \):
\[ y(0) = 0 + \sqrt{0} = 0 \]

2. Evaluate at \( x = 4 \):
\[ y(4) = 4 + \sqrt{4} = 4 + 2 = 6 \]

### Step 4: Identify the least and greatest values
Since the function is strictly increasing, the least value occurs at the left endpoint and the greatest value occurs at the right endpoint.

- Least value: \( y(0) = 0 \)
- Greatest value: \( y(4) = 6 \)

### Conclusion
Thus, the least value of the function \( y = x + \sqrt{x} \) on the interval \([0, 4]\) is \( 0 \) and the greatest value is \( 6 \).

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