The least value of `f (x) =x^3/3-abx` occurs at `x=`

Options

A. G.M. of a,b
B. A.M. of a,b
C. H.M of a,b
D. None of these

Answer

To find the least value of the function \( f(x) = \frac{x^3}{3} - abx \), we will follow these steps:

### Step 1: Differentiate the function
First, we need to find the first derivative of the function \( f(x) \).

\[ f'(x) = \frac{d}{dx} \left( \frac{x^3}{3} - abx \right) = x^2 - ab \]

### Step 2: Set the first derivative to zero
To find the critical points, we set the first derivative equal to zero.

\[ x^2 - ab = 0 \]

### Step 3: Solve for \( x \)
Now, we solve for \( x \).

\[ x^2 = ab \implies x = \pm \sqrt{ab} \]

### Step 4: Differentiate again to find the nature of critical points
Next, we need to find the second derivative to determine whether we have a minimum or maximum.

\[ f''(x) = \frac{d}{dx}(x^2 - ab) = 2x \]

### Step 5: Evaluate the second derivative at critical points
We will evaluate the second derivative at the critical points \( x = \sqrt{ab} \) and \( x = -\sqrt{ab} \).

For \( x = \sqrt{ab} \):

\[ f''(\sqrt{ab}) = 2\sqrt{ab} > 0 \]

This indicates that \( x = \sqrt{ab} \) is a point of local minimum.

For \( x = -\sqrt{ab} \):

\[ f''(-\sqrt{ab}) = -2\sqrt{ab} < 0 \]

This indicates that \( x = -\sqrt{ab} \) is a point of local maximum.

### Conclusion
The least value of \( f(x) \) occurs at:

\[ x = \sqrt{ab} \]