Question

F(x)=5(3x+x^3-4x^4)^3

Answer 1

$f'(x)=15(3x+x^3-4x^4)^2 (3+3x^2-16x^3)$

Answer 2

To analyze the function $F(x) = 5(3x + x^3 - 4x^4)^3$, we can break it down step-by-step.

Step 1: Identify the main expression within the parentheses. The main expression within the parentheses is $3x + x^3 - 4x^4$.

Step 2: Cube the main expression. Cubing the main expression gives us $(3x + x^3 - 4x^4)^3$.

Step 3: Multiply the cubed expression by 5. Multiplying the cubed expression by 5 gives us the final function $F(x) = 5(3x + x^3 - 4x^4)^3$.

Let's analyze the behavior of this function:

1. Degree of the function: The highest power of $x$ in the expression $3x + x^3 - 4x^4$ is $x^3$ (cubic term), and then when cubed, it becomes $x^9$ (since $(x^a)^b = x^{a \cdot b}$). Therefore, $F(x)$ is a polynomial of degree 9.

2. Leading Coefficient: The leading coefficient is the coefficient of the highest power term in the polynomial. In this case, it is $5$ since the highest power term is $x^9$.

3. Symmetry: The function $F(x)$ does not have any apparent symmetry.

4. Roots or Zeros: To find the roots or zeros of $F(x)$, we set $F(x) = 0$ and solve for $x$: $5(3x + x^3 - 4x^4)^3 = 0$

This equation has two solutions:

1. $3x + x^3 - 4x^4 = 0$
2. $x^3(3 - 4x^3 + x^2) = 0$

From here, we can find the specific values of $x$. However, the actual values will depend on the specific form of the quadratic equation $3 - 4x^3 + x^2$, which can only be determined by solving it directly.

5. Behavior as $x$ approaches positive infinity or negative infinity: As $x$ approaches positive or negative infinity, the term $(3x + x^3 - 4x^4)$ will dominate the function, and the function will tend to positive or negative infinity, respectively.

6. Critical points and Local Extrema: To find critical points, we need to find the values of $x$ that make the derivative of $F(x)$ equal to zero. Then, we can examine the second derivative to determine whether each critical point corresponds to a local maximum or minimum. However, finding the exact critical points will require further analysis of the derivative of $F(x)$ which is a cumbersome process.

7. Concavity: To determine the concavity of $F(x)$, we would need to analyze the sign of the second derivative of $F(x)$ at different intervals.

Overall, the function $F(x) = 5(3x + x^3 - 4x^4)^3$ is a polynomial of degree 9 with a leading coefficient of 5. It may have multiple real roots, local extrema, and concavity changes. For a more detailed analysis and specific numerical values, further calculations and graphing may be necessary.