`f(x)= {:{(x+asqrt2 sin x ,, 0 le x lt pi/4),( 2x cot x +b , , pi/4 le x le pi/2), (a cos 2 x - b sin x , , pi/2 lt x le pi):}`continuous function `AA x in [ 0,pi] " then " 5(a/b)^(2)` equals ....

Answer

To solve the problem, we need to ensure that the function \( f(x) \) is continuous at the points where the pieces of the function meet, specifically at \( x = \frac{\pi}{4} \) and \( x = \frac{\pi}{2} \).

### Step 1: Ensure continuity at \( x = \frac{\pi}{4} \)

We need to set the left-hand limit equal to the right-hand limit at \( x = \frac{\pi}{4} \):

\[ f\left(\frac{\pi}{4}^-\right) = f\left(\frac{\pi}{4}^+\right) \]

Calculating \( f\left(\frac{\pi}{4}^-\right) \):

\[ f\left(\frac{\pi}{4}^-\right) = a \sqrt{2} \sin\left(\frac{\pi}{4}\right) = a \sqrt{2} \cdot \frac{1}{\sqrt{2}} = a \]

Calculating \( f\left(\frac{\pi}{4}^+\right) \):

\[ f\left(\frac{\pi}{4}^+\right) = 2\left(\frac{\pi}{4}\right) \cot\left(\frac{\pi}{4}\right) + b = \frac{\pi}{2} + b \]

Setting them equal gives:

\[ a = \frac{\pi}{2} + b \quad \text{(1)} \]

### Step 2: Ensure continuity at \( x = \frac{\pi}{2} \)

Next, we set the left-hand limit equal to the right-hand limit at \( x = \frac{\pi}{2} \):

\[ f\left(\frac{\pi}{2}^-\right) = f\left(\frac{\pi}{2}^+\right) \]

Calculating \( f\left(\frac{\pi}{2}^-\right) \):

\[ f\left(\frac{\pi}{2}^-\right) = 2\left(\frac{\pi}{2}\right) \cot\left(\frac{\pi}{2}\right) + b = 0 + b = b \]

Calculating \( f\left(\frac{\pi}{2}^+\right) \):

\[ f\left(\frac{\pi}{2}^+\right) = a \cos(2 \cdot \frac{\pi}{2}) - b \sin\left(\frac{\pi}{2}\right) = a \cdot (-1) - b = -a - b \]

Setting them equal gives:

\[ b = -a - b \quad \Rightarrow \quad 2b = -a \quad \Rightarrow \quad a = -2b \quad \text{(2)} \]

### Step 3: Substitute equation (2) into equation (1)

Now we substitute \( a = -2b \) into equation (1):

\[ -2b = \frac{\pi}{2} + b \]

Rearranging gives:

\[ -3b = \frac{\pi}{2} \quad \Rightarrow \quad b = -\frac{\pi}{6} \]

### Step 4: Find \( a \)

Using \( b = -\frac{\pi}{6} \) in equation (2):

\[ a = -2\left(-\frac{\pi}{6}\right) = \frac{\pi}{3} \]

### Step 5: Calculate \( 5\left(\frac{a}{b}\right)^2 \)

Now we calculate \( \frac{a}{b} \):

\[ \frac{a}{b} = \frac{\frac{\pi}{3}}{-\frac{\pi}{6}} = -2 \]

Now, we find \( 5\left(\frac{a}{b}\right)^2 \):

\[ 5\left(-2\right)^2 = 5 \cdot 4 = 20 \]

### Final Answer

Thus, the value of \( 5\left(\frac{a}{b}\right)^2 \) is:

\[ \boxed{20} \]