If M and m are the greatest and least value of the function `f(x)=(cos^(-1)x)^2+(sin^(-1)x)^2` then the value of `((M+9m)/m)^3` is …..

Answer

To solve the problem, we need to find the greatest (M) and least (m) values of the function \( f(x) = (\cos^{-1} x)^2 + (\sin^{-1} x)^2 \) and then compute the value of \( \left( \frac{M + 9m}{m} \right)^3 \).

### Step 1: Understanding the function
We know that:
\[ \cos^{-1} x + \sin^{-1} x = \frac{\pi}{2} \]
This means we can express \( \cos^{-1} x \) in terms of \( \sin^{-1} x \):
\[ \cos^{-1} x = \frac{\pi}{2} - \sin^{-1} x \]

### Step 2: Substitute into the function
Substituting this into \( f(x) \):
\[ f(x) = \left( \frac{\pi}{2} - \sin^{-1} x \right)^2 + (\sin^{-1} x)^2 \]

### Step 3: Expand the function
Expanding the square:
\[ f(x) = \left( \frac{\pi^2}{4} - \pi \sin^{-1} x + (\sin^{-1} x)^2 \right) + (\sin^{-1} x)^2 \]
\[ = \frac{\pi^2}{4} - \pi \sin^{-1} x + 2(\sin^{-1} x)^2 \]

### Step 4: Completing the square
To find the minimum and maximum values, we can complete the square for the quadratic in \( \sin^{-1} x \):
\[ f(x) = \frac{\pi^2}{4} - \pi \sin^{-1} x + 2(\sin^{-1} x)^2 \]
Let \( y = \sin^{-1} x \):
\[ f(y) = 2y^2 - \pi y + \frac{\pi^2}{4} \]

### Step 5: Finding the vertex
The vertex of a quadratic \( ay^2 + by + c \) occurs at \( y = -\frac{b}{2a} \):
\[ y = -\frac{-\pi}{2 \cdot 2} = \frac{\pi}{4} \]

### Step 6: Minimum value (m)
Substituting \( y = \frac{\pi}{4} \):
\[ m = f\left(\frac{\pi}{4}\right) = 2\left(\frac{\pi}{4}\right)^2 - \pi\left(\frac{\pi}{4}\right) + \frac{\pi^2}{4} \]
\[ = 2 \cdot \frac{\pi^2}{16} - \frac{\pi^2}{4} + \frac{\pi^2}{4} = \frac{\pi^2}{8} \]

### Step 7: Maximum value (M)
The maximum value occurs at the endpoints of the range of \( \sin^{-1} x \), which is from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).

Calculating \( f\left(-\frac{\pi}{2}\right) \):
\[ M = f\left(-\frac{\pi}{2}\right) = 2\left(-\frac{\pi}{2}\right)^2 - \pi\left(-\frac{\pi}{2}\right) + \frac{\pi^2}{4} \]
\[ = 2 \cdot \frac{\pi^2}{4} + \frac{\pi^2}{2} + \frac{\pi^2}{4} = \frac{5\pi^2}{4} \]

### Step 8: Calculate \( \frac{M + 9m}{m} \)
Now we compute:
\[ \frac{M + 9m}{m} = \frac{\frac{5\pi^2}{4} + 9 \cdot \frac{\pi^2}{8}}{\frac{\pi^2}{8}} \]
Finding a common denominator:
\[ = \frac{\frac{5\pi^2}{4} + \frac{9\pi^2}{8}}{\frac{\pi^2}{8}} = \frac{\frac{10\pi^2}{8} + \frac{9\pi^2}{8}}{\frac{\pi^2}{8}} = \frac{19\pi^2/8}{\pi^2/8} = 19 \]

### Step 9: Calculate \( \left( \frac{M + 9m}{m} \right)^3 \)
Finally:
\[ \left( \frac{M + 9m}{m} \right)^3 = 19^3 = 6859 \]

### Final Answer
The value of \( \left( \frac{M + 9m}{m} \right)^3 \) is \( 6859 \).

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