If `f(x) = int (x^(3) - 2x^(2) + 3)e^(3x)` dx then the number of critical points of f are

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the number of critical points of the function \( f(x) = \int (x^3 - 2x^2 + 3)e^{3x} \, dx \), we need to follow these steps:

### Step 1: Differentiate \( f(x) \)
To find the critical points, we first need to compute the derivative \( f'(x) \). By the Fundamental Theorem of Calculus, the derivative of an integral is simply the integrand evaluated at \( x \):

\[ f'(x) = (x^3 - 2x^2 + 3)e^{3x} \]

### Step 2: Set the derivative equal to zero
Next, we find the critical points by setting the derivative equal to zero:

\[ f'(x) = (x^3 - 2x^2 + 3)e^{3x} = 0 \]

Since \( e^{3x} \) is never zero for any real \( x \), we can simplify our equation to:

\[ x^3 - 2x^2 + 3 = 0 \]

### Step 3: Analyze the cubic equation
Now we need to analyze the cubic equation \( x^3 - 2x^2 + 3 = 0 \) to find its roots. We can check for rational roots using the Rational Root Theorem or simply test some values.

Testing \( x = -1 \):

\[ (-1)^3 - 2(-1)^2 + 3 = -1 - 2 + 3 = 0 \]

Thus, \( x = -1 \) is a root. We can now factor the cubic polynomial using \( x + 1 \) as a factor.

### Step 4: Factor the cubic polynomial
Using synthetic division or polynomial long division, we divide \( x^3 - 2x^2 + 3 \) by \( x + 1 \):

\[ x^3 - 2x^2 + 3 = (x + 1)(x^2 - 3x + 3) \]

### Step 5: Solve the quadratic equation
Now we need to solve the quadratic equation \( x^2 - 3x + 3 = 0 \) using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 3}}{2 \cdot 1} = \frac{3 \pm \sqrt{9 - 12}}{2} = \frac{3 \pm \sqrt{-3}}{2} \]

This gives us:

\[ x = \frac{3 \pm i\sqrt{3}}{2} \]

### Step 6: Determine the number of critical points
The quadratic equation yields two complex roots, while we already found one real root \( x = -1 \). Therefore, the only real critical point is \( x = -1 \).

### Conclusion
Thus, the number of critical points of the function \( f(x) \) is:

\[ \text{Number of critical points} = 1 \]

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