If the area bounded by `f(x)=sqrt(tan x), y=f(c), x=0 and x=a, 0ltcltalt(pi)/(2)` is minimum then find the value of c.

Explanation

To find the value of \( c \) that minimizes the area bounded by the curve \( f(x) = \sqrt{\tan x} \), the line \( y = f(c) \), and the vertical lines \( x = 0 \) and \( x = a \) (where \( 0 < c < a < \frac{\pi}{2} \)), we can follow these steps:

### Step 1: Define the Area
The area \( A \) bounded by the curve and the lines can be expressed as:
\[ A = \int_0^c \left( f(c) - f(x) \right) \, dx + \int_c^a \left( f(x) - f(c) \right) \, dx \]
Substituting \( f(x) = \sqrt{\tan x} \), we have:
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