If `f(x)={{:(x^(3)(1-x)",",xle0),(xlog_(e)x+3x",",xgt0):}` then which of the following is not true?

Options

A. f(x) has point of maxima at x = 0
B. f(x) has point minima at `x=e^(-4)`
C. f(x) has range R
D. none of these

Explanation

To solve the problem, we need to analyze the function \( f(x) \) defined in two parts:

1. For \( x \leq 0 \):
\[ f(x) = x^3(1 - x) \]

2. For \( x > 0 \):
\[ f(x) = x \log_e x + 3x \]

We need to determine which of the provided statements about this function is not true.

### Step 1: Find the derivative \( f'(x) \)

**For \( x \leq 0 \)**:
\[ f(x) = x^3(1 - x) = x^3 - x^4 \]
Differentiating:
\[ f'(x) = 3x^2 - 4x^3 \]

**For \( x > 0 \)**:
\[ f(x) = x \log_e x + 3x \]
Using the product rule and the derivative of \( \log_e x \):
\[ f'(x) = \log_e x + 1 + 3 = \log_e x + 4 \]

### Step 2: Analyze critical points

**For \( x \leq 0 \)**:
Set \( f'(x) = 0 \):
\[ 3x^2 - 4x^3 = 0 \]
Factoring out \( x^2 \):
\[ x^2(3 - 4x) = 0 \]
This gives \( x = 0 \) or \( x = \frac{3}{4} \) (but \( \frac{3}{4} \) is not in the domain \( x \leq 0 \)). Thus, the critical point is \( x = 0 \).

**For \( x > 0 \)**:
Set \( f'(x) = 0 \):
\[ \log_e x + 4 = 0 \implies \log_e x = -4 \implies x = e^{-4} \]

### Step 3: Determine the nature of critical points

**At \( x = 0 \)**:
- To check if it's a maximum or minimum, we can evaluate the second derivative or analyze the sign of \( f'(x) \) around \( x = 0 \).

**For \( x > 0 \)**:
- The function \( \log_e x + 4 \) is negative for \( x < e^{-4} \) and positive for \( x > e^{-4} \), indicating that \( x = e^{-4} \) is a minimum.

### Step 4: Conclusion on the statements

Now we need to evaluate the options given in the problem statement to find which one is not true.

1. \( f(x) \) has a point of maxima at \( x = 0 \) (True).
2. \( f(x) \) has a point of minima at \( x = e^{-4} \) (True).
3. The function is defined for all real \( x \) (False, since it is not defined for negative values of \( x \)).
4. The range of \( f(x) \) is all real numbers (False, since the logarithmic part restricts the range).

Based on this analysis, the statement that is not true is the one regarding the function being defined for all real \( x \).

### Final Answer:
The statement that is not true is: **The function is defined for all real \( x \).**