Try to find the value of a such that the function f(x)=asinx+ (1/3)
sin3x gets its extremum points at the point x= (π / 3).
Decide if it is maximum or minimum, and find it.
To find the value of 'a' such that the function f(x) = asinx + (1/3)sin3x has its extremum points at x = π / 3, we need to analyze the behavior of the function at that point.
Let's start by finding the derivative of f(x) with respect to x. The derivative will help us determine whether the extremum point is a maximum or minimum. We can use the chain rule for differentiating the term sin3x:
f'(x) = a(d/dx(sinx)) + (1/3)(d/dx(sin3x))
= acosx + (1/3)(3cos3x)
= acosx + cos3x
Now, to find the extremum point, we need to set the derivative equal to zero and solve for x:
acosx + cos3x = 0
At this point, it might be challenging to find an exact value of x that satisfies this equation. However, we can use numerical methods (e.g., graphing calculators or software) to estimate the value of x.
After finding the approximate value of x, you can substitute it back into the original function f(x) = asinx + (1/3)sin3x to find the corresponding value of f(x). Depending on whether f(x) has a positive or negative value at that point, we can determine if it is a maximum or minimum, respectively.
Remember that the value of 'a' remains undetermined as it was not mentioned in the question.