A function is defined as `f(x) = ax^(2) – b|x|` where a and b are constants then at x = 0 we will have a maxima of f(x) if

Options

A. `a gt 0 , b gt 0`
B. `a gt 0 , b lt 0`
C. `a lt 0 , b lt 0`
D. `a lt 0 , b gt 0`

Answer

To determine the conditions under which the function \( f(x) = ax^2 - b|x| \) has a maximum at \( x = 0 \), we will follow these steps:

### Step 1: Evaluate the function at \( x = 0 \)
First, we need to find the value of the function at \( x = 0 \):
\[ f(0) = a(0)^2 - b|0| = 0 \]

### Step 2: Find the first derivative of the function
Next, we find the first derivative of \( f(x) \). Since the function involves an absolute value, we will consider two cases: when \( x \geq 0 \) and \( x < 0 \).

1. For \( x \geq 0 \):
\[ f(x) = ax^2 - bx \]
\[ f'(x) = 2ax - b \]

2. For \( x < 0 \):
\[ f(x) = ax^2 + bx \]
\[ f'(x) = 2ax + b \]

### Step 3: Evaluate the first derivative at \( x = 0 \)
Now, we evaluate \( f'(x) \) at \( x = 0 \):
- For \( x \geq 0 \):
\[ f'(0) = 2a(0) - b = -b \]
- For \( x < 0 \):
\[ f'(0) = 2a(0) + b = b \]

For \( f(x) \) to have a maximum at \( x = 0 \), the first derivative must change sign around \( x = 0 \). This means:
\[ f'(0) = -b < 0 \quad \text{(for } x \geq 0\text{)} \]
\[ f'(0) = b > 0 \quad \text{(for } x < 0\text{)} \]

### Step 4: Determine the conditions for maxima
From the above conditions, we conclude that:
- For \( x \geq 0 \), \( -b < 0 \) implies \( b > 0 \).
- For \( x < 0 \), \( b > 0 \) is already established.

Thus, we have:
\[ b > 0 \]

### Step 5: Find the second derivative
To confirm that we have a maximum, we need to check the second derivative:
1. For \( x \geq 0 \):
\[ f''(x) = 2a \]
2. For \( x < 0 \):
\[ f''(x) = 2a \]

### Step 6: Conditions on \( a \)
For \( x = 0 \) to be a maximum, we need:
\[ f''(0) < 0 \implies 2a < 0 \implies a < 0 \]

### Conclusion
Thus, for \( f(x) = ax^2 - b|x| \) to have a maximum at \( x = 0 \), the conditions are:
\[ a < 0 \quad \text{and} \quad b > 0 \]