If f^''(x)>0, ∀x ∈R, f^'(3)=0 and g(x)=f(tan^2 x-2. tanx+4), 0<x<π/ 2, then g(x) is increasing i

step 1 In discussion, given second derivative of f of x greater than 0 for all x belongs to r and deriv

If f^''(x)>0, ∀x ∈R, f^'(3)=0 and g(x)=f(tan^2 x-2. tanx+4),...

If $f^{\prime \prime}(x)>0, \forall x \in R, f^{\prime}(3)=0$ and $g(x)=f\left(\tan ^{2} x-2\right.$ $\tan x+4), 0<x<\pi / 2$, then $g(x)$ is increasing in (A) $\left(0, \frac{\pi}{4}\right)$ (B) $\left(0, \frac{\pi}{4}\right) \cup\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$ (B) $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$ (D) $\left(0, \frac{\pi}{2}\right)$

If $f^{\prime \prime}(x)>0, \forall x \in R, f^{\prime}(3)=0$ and $g(x)=f\left(\tan ^{2} x-2\right.$
$\tan x+4), 0<x<\pi / 2$, then $g(x)$ is increasing in
(A) $\left(0, \frac{\pi}{4}\right)$
(B) $\left(0, \frac{\pi}{4}\right) \cup\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$
(B) $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$
(D) $\left(0, \frac{\pi}{2}\right)$

Key Concepts

Composite Function Analysis

When analyzing a composite function, it's important to consider both the outer and inner functions, as well as their individual properties. The overall behavior, such as increasing or decreasing trends, depends on how the derivative of the outer function interacts with the derivative of the inner function, typically through the application of the chain rule. This analysis is critical for understanding the dynamics of the composite function as a whole.

Chain Rule

The chain rule is a fundamental algorithm in calculus for differentiating composite functions. It states that if a function is given as the composition of two functions, say h(x) = f(g(x)), then the derivative of h(x) is the product of the derivative of the outer function evaluated at g(x) and the derivative of the inner function. This concept helps in breaking down the process of differentiation for complicated compositions into simpler parts.

Monotonicity via Derivatives

Monotonicity in a function refers to its behavior of either increasing or decreasing throughout an interval. A key insight is that if the derivative of a function is positive over an interval, the function is increasing on that interval. This derivative test for monotonicity is essential in determining how functions change and in identifying intervals where the function is increasing or decreasing.

Strict Convexity

A function is strictly convex if its second derivative is positive across its domain. This implies that the function curves upward and any line segment connecting two points on the graph lies above the graph itself. Strict convexity ensures that there is a unique minimum point and that the first derivative is strictly increasing, which is pivotal when studying the function’s behavior and response to changes in its argument.

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