Question

\( f(x)= \begin{cases} |x^3 + x^2 + 3x + \sin x|\left(3 + \sin\frac{1}{x}\right), & x\neq 0 \\ 0, & x=0 \end{cases} \) The number of points, where \( f(x) \) attains its minimum value, is

• A. 1
• B. 2
• C. 3
• D. infinitely many

Hint:

For absolute value expressions, minimum occurs when inside becomes zero.

Answer 1

The Correct Option is A

To determine the number of points at which the function \( f(x) \) attains its minimum value, we need to analyze the behavior of the given piecewise function:

\(f(x)= \begin{cases} |x^3 + x^2 + 3x + \sin x|\left(3 + \sin\frac{1}{x}\right), & x\neq 0 \\ 0, & x=0 \end{cases}\)

First, let's consider the case when \( x = 0 \):

• For \( x = 0 \), \(f(0) = 0\).

Next, consider the case when \( x \neq 0 \):

• The expression \(|x^3 + x^2 + 3x + \sin x|\) is the absolute value of a polynomial function plus the sine function, which can be zero or positive, depending on \( x \).
• The expression \(3 + \sin\frac{1}{x}\) is always positive as it is the sum of 3 and a sine function, and the sine function oscillates between -1 and 1.

To find where the function \(f(x)\) attains its minimum, note that:

• The minimum value of \(|x^3 + x^2 + 3x + \sin x|\left(3 + \sin\frac{1}{x}\right)\) for \( x \neq 0 \) can only occur when \(|x^3 + x^2 + 3x + \sin x| = 0\) because the other factor is always positive.
• Hence, we need \(x^3 + x^2 + 3x + \sin x = 0\).

The polynomial equation \(x^3 + x^2 + 3x + \sin x = 0\) cannot be zero if \(\sin x\) is negligible, as the polynomial part does not allow a zero value at any point except where \( x = 0 \). Therefore, for \( x \neq 0 \), \(f(x) \gt 0\) holds.

Thus, the minimum value of \( f(x) \) is 0, which is attained only at \( x = 0 \).

Therefore, the number of points where \( f(x) \) attains its minimum value is 1.