Let $$g(x) = f(x) + f(1-x)$$ and $$f''(x) > 0$$, $$x \in (0, 1)$$. If $$g$$ is decreasing in the interval $$(0, \alpha)$$ and increasing in the interval $$(\alpha, 1)$$, then $$\tan^{-1}(2\alpha) + \tan^{-1}\left(\frac{1}{\alpha}\right) + \tan^{-1}\left(\frac{\alpha+1}{\alpha}\right)$$ is equal to

To solve this problem, we need to determine the value of $$\alpha$$ using the properties of the function $$g(x)$$ and then evaluate the inverse trigonometric expression.

Given:

$$g(x) = f(x) + f(1 - x)$$

To find the intervals of increase and decrease, we take the first derivative:

$$g'(x) = f'(x) + f'(1 - x) \cdot (-1) = f'(x) - f'(1 - x)$$

We are told $$g(x)$$ changes from decreasing to increasing at $$x = \alpha$$. This implies that $$x = \alpha$$ is a critical point where $$g'(\alpha) = 0$$:

$$f'(\alpha) - f'(1 - \alpha) = 0 \implies f'(\alpha) = f'(1 - \alpha)$$

Since $$f''(x) > 0$$ for $$x \in (0, 1)$$, the function $$f'(x)$$ is strictly increasing. In a strictly increasing function, if $$f'(a) = f'(b)$$, then $$a = b$$.

$$\alpha = 1 - \alpha \implies 2\alpha = 1 \implies \alpha = \frac{1}{2}$$

Now we substitute $$\alpha = \frac{1}{2}$$ into the required expression:

$$E = \tan^{-1}(2\alpha) + \tan^{-1}\left(\frac{1}{\alpha}\right) + \tan^{-1}\left(\frac{\alpha+1}{\alpha}\right)$$

Substitute $$\alpha = \frac{1}{2}$$:

• $$2\alpha = 2(1/2) = 1$$
• $$1/\alpha = 1/(1/2) = 2$$
• $$(\alpha+1)/\alpha = (1/2 + 1) / (1/2) = (3/2) / (1/2) = 3$$

The expression becomes:

$$E = \tan^{-1}(1) + \tan^{-1}(2) + \tan^{-1}(3)$$

We know $$\tan^{-1}(1) = \frac{\pi}{4}$$. For the remaining terms, use the identity $$\tan^{-1}x + \tan^{-1}y = \pi + \tan^{-1}\left(\frac{x+y}{1-xy}\right)$$ (since $$xy > 1$$):

$$\tan^{-1}(2) + \tan^{-1}(3) = \pi + \tan^{-1}\left(\frac{2+3}{1 - (2 \cdot 3)}\right)$$

$$\tan^{-1}(2) + \tan^{-1}(3) = \pi + \tan^{-1}\left(\frac{5}{-5}\right)$$

$$\tan^{-1}(2) + \tan^{-1}(3) = \pi + \tan^{-1}(-1)$$

$$\tan^{-1}(2) + \tan^{-1}(3) = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$$

Adding the first term back:

$$E = \frac{\pi}{4} + \frac{3\pi}{4} = \pi$$

Final Answer: A ($$\pi$$)