If f(x)=`x^(3)-kx^(2)+2x` and f(-x)= -f(x), the value of k is_

Options

A. -2
B. 0
C. 1
D. 2

Answer

To find the value of \( k \) in the function \( f(x) = x^3 - kx^2 + 2x \) given that \( f(-x) = -f(x) \), we can follow these steps:

### Step 1: Write down the function \( f(x) \)
We have:
\[ f(x) = x^3 - kx^2 + 2x \]

### Step 2: Calculate \( f(-x) \)
Substituting \(-x\) into the function:
\[ f(-x) = (-x)^3 - k(-x)^2 + 2(-x) \]
This simplifies to:
\[ f(-x) = -x^3 - kx^2 - 2x \]

### Step 3: Write down the expression for \(-f(x)\)
Now we calculate \(-f(x)\):
\[ -f(x) = - (x^3 - kx^2 + 2x) = -x^3 + kx^2 - 2x \]

### Step 4: Set \( f(-x) \) equal to \(-f(x)\)
According to the problem, we have:
\[ f(-x) = -f(x) \]
Thus, we equate the two expressions:
\[ -x^3 - kx^2 - 2x = -x^3 + kx^2 - 2x \]

### Step 5: Simplify the equation
Removing \(-x^3\) and \(-2x\) from both sides gives:
\[ -kx^2 = kx^2 \]

### Step 6: Combine like terms
Adding \( kx^2 \) to both sides results in:
\[ -kx^2 - kx^2 = 0 \]
This simplifies to:
\[ -2kx^2 = 0 \]

### Step 7: Solve for \( k \)
Since \( x^2 \) cannot be zero for all \( x \), we must have:
\[ -2k = 0 \implies k = 0 \]

### Conclusion
The value of \( k \) is:
\[ \boxed{0} \]