Answer is: option3

Solution:

To find the point of inflection of the function:

\[ f(x) = \sqrt{x} + \frac{1}{\sqrt{x}} \] we need to compute the second derivative and find where it changes sign.

Rewriting the function:

\[ f(x) = x^{1/2} + x^{-1/2} \]

Differentiate:

\[ f'(x) = \frac{1}{2} x^{-1/2} - \frac{1}{2} x^{-3/2} \] \[ f'(x) = \frac{1}{2\sqrt{x}} - \frac{1}{2x^{3/2}} \]

Differentiate again:

\[ f''(x) = -\frac{1}{4} x^{-3/2} + \frac{3}{4} x^{-5/2} \] \[ f''(x) = -\frac{1}{4x^{3/2}} + \frac{3}{4x^{5/2}} \]

Finding the point of inflection:

A point of inflection occurs where \( f''(x) = 0 \), so:

\[ -\frac{1}{4x^{3/2}} + \frac{3}{4x^{5/2}} = 0 \]

Factor out \( \frac{1}{4x^{5/2}} \):

\[ \frac{1}{4x^{5/2}} (-x + 3) = 0 \]

Since \( \frac{1}{4x^{5/2}} \neq 0 \), we solve:

\[ -x + 3 = 0 \] \[ x = 3 \]

Confirming the inflection point:

To confirm an inflection point, check the sign of \( f''(x) \) on either side of \( x = 3 \):

• For \( x < 3 \) (e.g., \( x = 2 \)), compute \( f''(2) \):

\[ f''(2) = -\frac{1}{4(2)^{3/2}} + \frac{3}{4(2)^{5/2}} \]

This gives a negative value.

• For \( x > 3 \) (e.g., \( x = 4 \)), compute \( f''(4) \):

\[ f''(4) = -\frac{1}{4(4)^{3/2}} + \frac{3}{4(4)^{5/2}} \]

This gives a positive value.

Since \( f''(x) \) changes sign at \( x = 3 \), there is a point of inflection at \( x = 3 \).