For a ∈[π, 2 π] and n ∈Z, the critical points of f(x)= (1)/(3) sina tan^3 x+(sina-1) tanx+√((a-2

step 1 is given and we have to find this critical point and it is given that alpha belongs to fire to t

For a ∈[π, 2 π] and n ∈Z, the critical points of f(x)=...

For $a \in[\pi, 2 \pi]$ and $n \in Z$, the critical points of $f(x)=$ $\frac{1}{3} \sin a \tan ^{3} x+(\sin a-1) \tan x+\sqrt{\frac{a-2}{8-a}}$ are
(A) $x=n \pi$
(B) $x=2 n \pi$
(C) $x=(2 n+1) \pi$
(D) None of these

Key Concepts

Solving Trigonometric Equations

Finding the critical points often involves setting the derivative equal to zero and solving the resulting trigonometric equation. This process requires an understanding of algebraic manipulation and the application of trigonometric identities to isolate and determine the variable values that satisfy the equation.

Trigonometric Functions and Their Properties

Trigonometric functions, such as the tangent function, have specific periodicity and discontinuity properties that are vital to understanding their behavior in different intervals. Knowledge of these properties is important when determining solution sets for equations derived from their derivatives, as these characteristics influence the existence and location of critical points.

Differentiation of Composite Functions

Differentiating composite functions, especially those involving trigonometric functions like the tangent function, requires the use of the chain rule. This rule facilitates computing the derivative of a function that is applied to another function, which is essential when dealing with expressions where multiple operations are nested.

Critical Points

A critical point of a function is where its derivative is either zero or undefined. These points are of central importance in calculus because they can indicate local maximums, minimums, or points of inflection. The concept is crucial in optimization and in understanding a function’s behavior.

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