The function f(x)=sinx cos^2 x has extremum at (a) x=π/ 2 (b) x=cos^-1(v √(3)) (c) x=cos^-1(-√(2

step 1 To find the extremum of a function, we find its derivative F dash of x and we equate that to 0.

The function f(x)=sinx cos^2 x has extremum at (a) x=π/ 2...

Key Concepts

Finding Extremum Using Derivatives

In calculus, one of the primary methods for determining the maximum or minimum values (extrema) of a function is to compute its derivative and set it equal to zero. This process identifies the critical points of the function, where the function’s slope is zero or undefined, which are the candidates for local extrema. Further tests, such as the first or second derivative test, can then be applied to classify these critical points as either minima or maxima.

Critical Points

Critical points are the values of the independent variable at which the derivative of a function is zero or does not exist. These points are fundamental in the study of calculus as they often correspond to the local maximum or minimum values of a function. By analyzing the behavior of the function around these points, one can determine whether they are points of local or global extremum.

Derivative of Trigonometric Functions

Derivatives of trigonometric functions form an essential part of differential calculus, especially when dealing with functions that involve sine, cosine, and other trigonometric expressions. Using rules like the product rule and chain rule, one can find the derivative of a function composed of trigonometric functions. This is crucial in analyzing the rate of change of the function and subsequently in finding its critical points and extrema.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to determine the angle whose trigonometric function equals a given value. They are essential in solving trigonometric equations, particularly when the solution involves identifying the specific angles at which a trigonometric condition holds. Understanding the domain and range of these functions is key to ensuring that the solutions are valid within the context of the problem.

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