If at each point of the curve `y=x^3-a x^2+x+1,` the tangent is inclined at an acute angle with the positive direction of the x-axis, then (a)`a >0` (b) `a<-sqrt(3)` (c)`-sqrt(3)< a< ` `sqrt(3)` (d) `non eoft h e s e`

Options

A. `agt0`
B. `alesqrt3`
C. `-sqrt3lealesqrt3`
D. none of these

Answer

To solve the problem, we need to analyze the curve given by the equation:

\[ y = x^3 - ax^2 + x + 1 \]

We want to find the conditions under which the tangent to this curve at any point makes an acute angle with the positive direction of the x-axis.

### Step 1: Find the derivative of the curve

The slope of the tangent to the curve at any point is given by the derivative \( \frac{dy}{dx} \). We will differentiate the equation with respect to \( x \):

\[ \frac{dy}{dx} = \frac{d}{dx}(x^3 - ax^2 + x + 1) \]

Using the power rule of differentiation:

\[ \frac{dy}{dx} = 3x^2 - 2ax + 1 \]

### Step 2: Set the condition for the slope

For the tangent to make an acute angle with the positive x-axis, the slope must be positive:

\[ 3x^2 - 2ax + 1 > 0 \]

### Step 3: Analyze the quadratic inequality

The expression \( 3x^2 - 2ax + 1 \) is a quadratic in \( x \). For this quadratic to be positive for all \( x \) (i.e., for all points on the curve), its discriminant must be less than zero. The discriminant \( D \) of a quadratic \( Ax^2 + Bx + C \) is given by:

\[ D = B^2 - 4AC \]

Here, \( A = 3 \), \( B = -2a \), and \( C = 1 \). Thus, the discriminant is:

\[ D = (-2a)^2 - 4 \cdot 3 \cdot 1 \]
\[ D = 4a^2 - 12 \]

### Step 4: Set the discriminant less than zero

For the quadratic to be positive for all \( x \):

\[ 4a^2 - 12 < 0 \]

### Step 5: Solve the inequality

Rearranging gives:

\[ 4a^2 < 12 \]
\[ a^2 < 3 \]

Taking the square root of both sides, we find:

\[ -\sqrt{3} < a < \sqrt{3} \]

### Conclusion

The range of \( a \) for which the tangent to the curve makes an acute angle with the positive direction of the x-axis is:

\[ -\sqrt{3} < a < \sqrt{3} \]

Thus, the correct option is:

(c) \(-\sqrt{3} < a < \sqrt{3}\)