Question

what is the max. value of the function y=sin6x in [0,Π]​

Answer 1

Answer:

The maximum & minimum value of

$y= sin6x \: in \: the \: given \: region \: in \: 1 \: \: and \: \: </strong><strong>-</strong><strong> </strong><strong>1$

The given region [0,Π ] is 1st & 2nd quadrant when the given function is non-negative

So the maximum value is 1

Answer 2

Answer:

Hey mate ! Your answer is at the bottom of my answer.

Step-by-step explanation:

Find the first derivative of the function.

Differentiate using the chain rule we get,

6cos(6x)

Find the second derivative of the function.

To find the local maximum and minimum values of the function, set the derivative equal to  0  and solve.

6cos(6x)=0

Take the inverse cosine of both sides of the equation to extract  x  from inside the cosine.

6x=arccos(0)

W.K.T. The exact value of arccos(0)  is  π  / 2

Therefore,

⇒6x  =  π  / 2

Divide each term by  6  and simplify.

⇒x  =  π  / 12

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from 2π  to find the solution in the fourth quadrant.

6

x=2π−π/2

Simplify the expression to find the second solution.

x=π  / 4

The solution to the equation  

6x=π/ 2.

x=π  / 12,π  / 4

Evaluate the second derivative at  x=π  / 12.

If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

−36sin(6(π/12))

Evaluate the second derivative.

−36

x=π/12  is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

x=π/12  is a local maximum

Find the y-value when  x=π  / 12.

y=1

Evaluate the second derivative at  x=π/4. If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

−36sin(6(π/4))

Evaluate the second derivative.

36x=π/4  is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

x=π/4  is a local minimum

Find the y-value when  x=π/4.

y=−1

These are the local extrema for  f(x)=sin(6x).

(π/12,1)  is a local maxima

(π/4,−1)  is a local minima

(π/12,1)  is the max. value.

Answer is 1 is the maximum value and -1 is the minimum value.

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