The shortest distance between the point `((3)/(2),0)` and the curve `y=sqrt(x),(x gt 0)`, is

Options

A. `(3)/(2)`
B. `(5)/(4)`
C. `(sqrt(3))/(2)`
D. `(sqrt(5))/(2)`

Explanation

To find the shortest distance between the point \((\frac{3}{2}, 0)\) and the curve \(y = \sqrt{x}\) (for \(x > 0\)), we can follow these steps:

### Step-by-Step Solution:

1. **Identify a point on the curve**:
Let’s denote a point on the curve as \(P(h, \sqrt{h})\), where \(h\) is a variable representing the x-coordinate of the point on the curve.

2. **Use the distance formula**:
The distance \(d\) between the point \((\frac{3}{2}, 0)\) and the point \(P(h, \sqrt{h})\) is given by the distance formula:
\[ d = \sqrt{(h - \frac{3}{2})^2 + (\sqrt{h} - 0)^2} \]

3. **Simplify the distance squared**:
To minimize the distance, we can minimize \(d^2\) instead:
\[ d^2 = (h - \frac{3}{2})^2 + (\sqrt{h})^2 \]
Expanding this gives:
\[ d^2 = (h - \frac{3}{2})^2 + h \]
\[ = (h^2 - 3h + \frac{9}{4}) + h \]
\[ = h^2 - 2h + \frac{9}{4} \]

4. **Complete the square**:
To make it easier to find the minimum, we complete the square:
\[ d^2 = (h^2 - 2h + 1) + \frac{9}{4} - 1 \]
\[ = (h - 1)^2 + \frac{5}{4} \]

5. **Find the minimum value**:
The term \((h - 1)^2\) is always non-negative and achieves its minimum value of \(0\) when \(h = 1\). Thus, the minimum value of \(d^2\) is:
\[ d^2_{\text{min}} = 0 + \frac{5}{4} = \frac{5}{4} \]

6. **Calculate the minimum distance**:
Therefore, the minimum distance \(d\) is:
\[ d_{\text{min}} = \sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2} \]

### Final Answer:
The shortest distance between the point \((\frac{3}{2}, 0)\) and the curve \(y = \sqrt{x}\) is \(\frac{\sqrt{5}}{2}\).

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