For what value of a,`f(x)=-x^3+4ax^2+2x-5` decreasing for all x .

Options

A. (1,2)
B. (3,4)
C. R
D. no value of a

Answer

To determine the value of \( a \) for which the function \( f(x) = -x^3 + 4ax^2 + 2x - 5 \) is decreasing for all \( x \), we will follow these steps:

### Step 1: Differentiate the function
We start by finding the derivative of the function \( f(x) \).

\[ f'(x) = \frac{d}{dx}(-x^3 + 4ax^2 + 2x - 5) \]

Calculating the derivative, we get:

\[ f'(x) = -3x^2 + 8ax + 2 \]

### Step 2: Set the condition for decreasing function
For the function to be decreasing for all \( x \), the derivative \( f'(x) \) must be less than or equal to zero for all \( x \).

\[ -3x^2 + 8ax + 2 < 0 \quad \text{for all } x \]

### Step 3: Analyze the quadratic expression
The expression \( -3x^2 + 8ax + 2 \) is a quadratic function in \( x \). For this quadratic to be negative for all \( x \), the following conditions must hold:

1. The coefficient of \( x^2 \) (which is -3) must be negative (which it is).
2. The discriminant \( D \) of the quadratic must be less than 0.

### Step 4: Calculate the discriminant
The discriminant \( D \) of the quadratic \( -3x^2 + 8ax + 2 \) is given by:

\[ D = b^2 - 4ac \]

Here, \( a = -3 \), \( b = 8a \), and \( c = 2 \):

\[ D = (8a)^2 - 4(-3)(2) \]
\[ D = 64a^2 + 24 \]

### Step 5: Set the discriminant less than zero
For the quadratic to be negative for all \( x \), we need:

\[ 64a^2 + 24 < 0 \]

### Step 6: Solve the inequality
However, \( 64a^2 + 24 \) is always positive for all real values of \( a \) because \( 64a^2 \) is non-negative and 24 is positive. Thus, there are no values of \( a \) that satisfy this inequality.

### Conclusion
Since the discriminant cannot be less than zero, there is no value of \( a \) for which the function \( f(x) \) is decreasing for all \( x \).