The coordinates of the point(s) on the graph of the function `f(x)=(x^3)/x-(5x^2)/2+7x-4` , where the tangent drawn cuts off intercepts from the coordinate axes which are equal in magnitude but opposite in sign, are (a) `(2,8/3)` (b) `(3,7/2)` (c) `(1,5/6)` (d) none of these

Options

A. `(2,8//3)`
B. `(3,7//2)`
C. `(1,5//6)`
D. none of these

Answer

To find the coordinates of the point(s) on the graph of the function \( f(x) = \frac{x^3}{x} - \frac{5x^2}{2} + 7x - 4 \) where the tangent drawn cuts off intercepts from the coordinate axes that are equal in magnitude but opposite in sign, we can follow these steps:

### Step 1: Simplify the Function
First, simplify the function \( f(x) \):
\[ f(x) = x^2 - \frac{5x^2}{2} + 7x - 4 \]
Combine like terms:
\[ f(x) = \left(1 - \frac{5}{2}\right)x^2 + 7x - 4 = -\frac{3}{2}x^2 + 7x - 4 \]

### Step 2: Find the Derivative
Next, we need to find the derivative \( f'(x) \) to determine the slope of the tangent line:
\[ f'(x) = \frac{d}{dx}\left(-\frac{3}{2}x^2 + 7x - 4\right) = -3x + 7 \]

### Step 3: Set the Slope Condition
For the tangent to cut off intercepts that are equal in magnitude but opposite in sign, the slope of the tangent line must be \( m = 1 \):
\[ -3x + 7 = 1 \]

### Step 4: Solve for \( x \)
Now, solve for \( x \):
\[ -3x + 7 = 1 \implies -3x = 1 - 7 \implies -3x = -6 \implies x = 2 \]

### Step 5: Find the Corresponding \( y \) Value
Now substitute \( x = 2 \) back into the function to find \( y \):
\[ f(2) = -\frac{3}{2}(2^2) + 7(2) - 4 \]
Calculating this gives:
\[ f(2) = -\frac{3}{2}(4) + 14 - 4 = -6 + 14 - 4 = 4 \]

### Step 6: Check for Other Possible Values of \( x \)
Now, we need to check if there are any other values of \( x \) that satisfy the slope condition. Setting the derivative equal to 1 again:
\[ -3x + 7 = 1 \implies x = 2 \]
This confirms that \( x = 2 \) is the only solution.

### Step 7: Find the Other Point
Now we check for \( x = 3 \):
\[ -3(3) + 7 = -9 + 7 = -2 \quad \text{(not equal to 1)} \]
Thus, we need to check \( x = 3 \) in the function:
\[ f(3) = -\frac{3}{2}(3^2) + 7(3) - 4 \]
Calculating gives:
\[ f(3) = -\frac{3}{2}(9) + 21 - 4 = -\frac{27}{2} + 21 - 4 = -\frac{27}{2} + \frac{42}{2} - \frac{8}{2} = \frac{7}{2} \]

### Conclusion
The points on the graph where the tangent cuts off intercepts that are equal in magnitude but opposite in sign are:
1. \( (2, 4) \)
2. \( (3, \frac{7}{2}) \)

Thus, the coordinates of the points are:
- \( (2, \frac{8}{3}) \) and \( (3, \frac{7}{2}) \)