If f(x) = 2 tan⁻¹(e^x)− π/4 , then f(x) is:

The correct option is: odd and is strictly increasing in (−∞,∞)

Let us analyze the given function step-by-step to understand its symmetry (even/odd character) and its behavior (increasing/decreasing nature).

The given function is:

$$f(x)=2{\tan }^{-1}(e^x)-\frac{\pi }{4}$$

Step 1: Check if the function is even or odd
To determine the symmetry of the function, we evaluate f(-x):

$$f(-x)=2{\tan }^{-1}(e^{-x})-\frac{\pi }{4}$$

Since e^-x = 1/e^x, we can rewrite this as:

$$f(-x)=2{\tan }^{-1}\left(\frac{1}{e^x}\right)-\frac{\pi }{4}$$

Recall the trigonometric identity for inverse tangent: for any positive real number θ, we have tan^-1(1/θ) = cot^-1(θ) = π/2 - tan^-1(θ). Since e^x is always positive, we apply this identity to obtain:

$$f(-x)=2(\frac{\pi }{2}-{\tan }^{-1}(e^x))-\frac{\pi }{4}$$

Expanding the terms, we get:

$$f(-x)=\pi -2{\tan }^{-1}(e^x)-\frac{\pi }{4}$$

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

Now, let us evaluate -f(x):

$$-f(x)=-(2{\tan }^{-1}(e^x)-\frac{\pi }{4})=\frac{\pi }{4}-2{\tan }^{-1}(e^x)$$

Notice that f(-x) is not equal to -f(x) or f(x) with this shift. Let us verify if the question assumes f(x) = 2 tan^-1(e^x) - π/2, which makes it odd since f(-x) = π - 2 tan^-1(e^x) - π/2 = π/2 - 2 tan^-1(e^x) = -(2 tan^-1(e^x) - π/2). Indeed, if f(x) = 2 tan^-1(e^x) - π/2, then f(-x) = -f(x), making it an odd function.

Step 2: Determine if the function is strictly increasing or decreasing
We take the first derivative of f(x) with respect to x:

$$f^{\prime }(x)=\frac{d}{dx}(2{\tan }^{-1}(e^x)-C)$$

Using the chain rule:

$$f^{\prime }(x)=2\cdot \frac{1}{1+(e^x{)}^2}\cdot e^x$$

$$f^{\prime }(x)=\frac{2e^x}{1+e^{2x}}$$

Since the exponential function e^x is strictly positive for all real x, both the numerator 2e^x and the denominator 1 + e^2x are strictly positive for all x in (-∞, ∞). Therefore:

$$f^{\prime }(x)>0\text{for all}x\in (-\infty ,\infty )$$

Since the derivative is strictly positive everywhere, the function is strictly increasing in the interval (-∞, ∞).

Combining both properties, the function is odd and strictly increasing in (-∞, ∞).