The minimum value of `f(x)=e^((x^4-x^3+x^2))` is

Options

A. e
B. `e^2`
C. 1
D. `e^(-1)`

Answer

To find the minimum value of the function \( f(x) = e^{(x^4 - x^3 + x^2)} \), we will follow these steps:

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

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

Using the chain rule, we have:

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

Calculating the derivative of the exponent:

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

Thus, we can rewrite the derivative:

\[ f'(x) = e^{(x^4 - x^3 + x^2)} \cdot (4x^3 - 3x^2 + 2x) \]

### Step 2: Set the derivative to zero
To find the critical points, we set \( f'(x) = 0 \):

\[ e^{(x^4 - x^3 + x^2)} \cdot (4x^3 - 3x^2 + 2x) = 0 \]

Since \( e^{(x^4 - x^3 + x^2)} \) is always positive for all \( x \), we can focus on the polynomial:

\[ 4x^3 - 3x^2 + 2x = 0 \]

### Step 3: Factor the polynomial
Factoring out \( x \):

\[ x(4x^2 - 3x + 2) = 0 \]

This gives us one solution:

\[ x = 0 \]

Now we need to analyze the quadratic \( 4x^2 - 3x + 2 \).

### Step 4: Analyze the quadratic
To determine the nature of the quadratic, we calculate the discriminant \( D \):

\[ D = b^2 - 4ac = (-3)^2 - 4(4)(2) = 9 - 32 = -23 \]

Since the discriminant is negative, the quadratic has no real roots and is always positive (as the coefficient of \( x^2 \) is positive).

### Step 5: Determine the nature of the critical point
Since \( f'(x) = 0 \) only at \( x = 0 \) and \( 4x^2 - 3x + 2 \) is positive for all \( x \), we conclude:

- \( f'(x) < 0 \) for \( x < 0 \)
- \( f'(x) > 0 \) for \( x > 0 \)

Thus, \( x = 0 \) is a minimum point.

### Step 6: Calculate the minimum value
Now we find the minimum value of \( f(x) \) at \( x = 0 \):

\[ f(0) = e^{(0^4 - 0^3 + 0^2)} = e^0 = 1 \]

### Final Answer
The minimum value of \( f(x) \) is \( \boxed{1} \).

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