Find the critical numbers of the function. (Enter your answers as a comma-separated list. Use n to denote any arbitrary integer values. If an answer does not exist, enter DNE.) f(θ)=6cosθ+3sin2θ g

The critical value of   are given by or ,

Step-by-step explanation:

The function to be evaluated is , the first derivative of the function must be taken in order to determine the set of critical numbers. Each derivative are found by using the differentiation rule for a sum of functions and rule of chain and subsequently simplified by trigonometric and algebraic means:

First derivative

The procedure to determine the critical number of the given function are described briefly:

1) First derivative is equalised to zero.

2) The resultant equation is solved.

Then,

Whose roots are:

and

The sine function is a continuous function with a range between 1 and -1, so, only the first root offers a realistic solution. In addition, such function is positive at first and second quadrants and has a periodicity of radians, the family of critical values are determined by the unse of inverse trigonometric functions:

There are two subsets of solutions:

or ,

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step-by-step explanation:

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