The function `f(x)=tanx+(1)/(x), AA x in (0, (pi)/(2))` has

Options

A. one local maximum
B. one local minimum
C. one local maximum and one minimum
D. no local maximum of minimum

Answer

To solve the problem, we need to analyze the function \( f(x) = \tan x + \frac{1}{x} \) for \( x \) in the interval \( (0, \frac{\pi}{2}) \).

### Step 1: Find the derivative of the function
We start by finding the derivative \( f'(x) \) of the function \( f(x) \).

\[ f'(x) = \frac{d}{dx}(\tan x) + \frac{d}{dx}\left(\frac{1}{x}\right) \]

Using the derivatives of \( \tan x \) and \( \frac{1}{x} \):

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

### Step 2: Set the derivative to zero
To find critical points, we set the derivative equal to zero:

\[ \sec^2 x - \frac{1}{x^2} = 0 \]

This can be rearranged to:

\[ \sec^2 x = \frac{1}{x^2} \]

### Step 3: Rewrite in terms of sine and cosine
Recall that \( \sec x = \frac{1}{\cos x} \), so we can rewrite \( \sec^2 x \):

\[ \frac{1}{\cos^2 x} = \frac{1}{x^2} \]

Cross-multiplying gives:

\[ x^2 = \cos^2 x \]

### Step 4: Analyze the function
We need to analyze the behavior of \( f'(x) \) in the interval \( (0, \frac{\pi}{2}) \).

1. As \( x \to 0 \), \( \tan x \to 0 \) and \( \frac{1}{x} \to \infty \), thus \( f(x) \to \infty \).
2. As \( x \to \frac{\pi}{2} \), \( \tan x \to \infty \) and \( \frac{1}{x} \to \frac{2}{\pi} \), thus \( f(x) \to \infty \).

### Step 5: Determine the nature of critical points
To determine whether the critical point found is a minimum or maximum, we can analyze the sign of \( f'(x) \):

- For \( x \) close to 0, \( \sec^2 x \) is large and positive, while \( -\frac{1}{x^2} \) is negative. Thus, \( f'(x) > 0 \).
- For \( x \) approaching \( \frac{\pi}{2} \), \( \sec^2 x \) becomes very large, and \( -\frac{1}{x^2} \) becomes less significant. Thus, \( f'(x) > 0 \).

Since \( f'(x) \) changes from positive to negative, there is a local minimum in the interval.

### Conclusion
The function \( f(x) = \tan x + \frac{1}{x} \) has **one local minimum** in the interval \( (0, \frac{\pi}{2}) \).