Which one of the following statements is correct in respect of the function f(x) = x3sinx?a)It has local maximum at x =0.b)It has local minimum at x = 0.c)It has neither maximum nor minimum at x =0.d)It has maximum value 1.Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Local Maximum and Minimum of a Function

To determine whether the function f(x) = x^3sin(x) has a local maximum or minimum at x = 0, we need to analyze the behavior of the function in the vicinity of this point.

1. Derivative Test:
The first step is to find the derivative of the function f(x). Using the product rule and chain rule, we can calculate:

f'(x) = (3x^2sin(x)) + (x^3cos(x))

2. Critical Points:
To find the critical points, we need to find the values of x for which f'(x) = 0 or f'(x) is undefined. In this case, f'(x) is defined for all values of x, so we only need to find the values of x for which f'(x) = 0.

Setting f'(x) = 0, we have:

(3x^2sin(x)) + (x^3cos(x)) = 0

3. Analysis of f'(x) = 0:
To analyze the behavior of f'(x) = 0, we need to consider the sign of f'(x) on either side of x = 0.

- For x < />
If we choose a negative value for x, f'(x) becomes negative. This is because sin(x) is negative in the third and fourth quadrants, and cos(x) is negative in the second and third quadrants. The negative terms dominate, resulting in a negative value for f'(x).

- For x > 0:
If we choose a positive value for x, f'(x) becomes positive. This is because sin(x) is positive in the first and second quadrants, and cos(x) is negative in the second and third quadrants. The positive terms dominate, resulting in a positive value for f'(x).

4. Conclusion:
Based on the analysis of f'(x) on either side of x = 0, we can conclude the following:

- Since f'(x) changes sign from negative to positive, x = 0 is a point of local minimum.
- Therefore, the correct statement is option 'b': It has a local minimum at x = 0.

Since the given statement is incorrect, the correct option is 'c': It has neither a maximum nor a minimum at x = 0.

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