Question

If \(f(x)=1-x-x^{3}\), then all real values of \(x\) satisfying the inequality \(1-f(x)-f^{3}(x)>f(1-5 x)\) are (1) \((-2,0) \cup(2, \infty)\) (2) \((-\infty,-2)\) (3) \((1,2)\) (4) \((0,2)\)

Short answer

(0, 2)

Step-by-step solution

Review the given function

The function is given by \[f(x) = 1 - x - x^3\].Let's denote this function as \(y\).

Rewrite the inequality

We need to solve the inequality: \[1 - f(x) - f^3(x) > f(1 - 5x)\].Substitute \(f(x) = 1 - x - x^3\) into the inequality.

Substitute for f(x) and simplify

Let's substitute \[f(x) = 1 - x - x^3\]and\[f^3(x) = (1 - x - x^3)^3\] into the inequality: \[1 - (1 - x - x^3) - (1 - x - x^3)^3 > f(1 - 5x)\].Simplify the left-hand side: \[x + x^3 - (1 - x - x^3)^3 > f(1 - 5x)\].

Find f(1 - 5x)

We need to find the function value for x:\[f(1 - 5x) = 1 - (1 - 5x) - (1 - 5x)^3\].Simplify:\[f(1 - 5x) = 1 - 1 + 5x - (1 - 15x + 75x^2 - 125x^3)\].Further simplification gives us:\[f(1 - 5x) = 5x - 1 + 15x - 75x^2 + 125x^3\].

Construct the simplified inequality

Using the simplified functions, rewrite the inequality: \[x + x^3 - (1 - x - x^3)^3 > 5x + 15x - 75x^2 + 125x^3\].Solve this inequality by substituting different intervals.

Test the intervals

By testing the intervals within the options given, verify which intervals satisfy the inequality. Check endpoints if needed.

Conclusion

From interval testing, the solution to the inequality is (0, 2).

Key concepts

inequality solving

Solving inequalities involves finding values of a variable that make the inequality true. Unlike solving equations, where an exact value is sought, inequalities often result in a range of values.

In the given exercise, we start with an inequality involving a function: \[1 - f(x) - f^3(x) > f(1 - 5x)\]. We first need to substitute the given function \[f(x) = 1 - x - x^3\]. This step transforms our problem into a more manageable expression that we can simplify and solve.

The key steps in solving any inequality include:

• Identifying the given inequality and any functions involved
• Substituting the functions and simplifying the expression
• Finding and testing intervals to determine where the inequality holds true

Understanding these steps will make inequality solving more approachable.

function substitution

Function substitution means replacing a function within a given expression or inequality with its equivalent expression.

In our exercise, we start by substituting \[f(x) = 1 - x - x^3\] into the inequality. After substitution, the inequality becomes: \[1 - (1 - x - x^3) - (1 - x - x^3)^3 > f(1 - 5x)\].

The purpose of function substitution is to transform the inequality into a more familiar or simpler form, often simplifying the process of solving it. Steps involved:

• Rewrite the given functions in their simplest form
• Replace every instance of the function in the inequality with this simpler form
• Simplify the resulting expression to make it easier to solve

This breaks down complex problems into easier parts.

polynomial simplification

Simplifying polynomials involves reducing them to a form that is easier to work with. This often includes combining like terms and reducing powers when possible.

In our problem, once we substitute the function \[f(x) = 1 - x - x^3\], we get: 1 - (1 - x - x^3) - (1 - x - x^3)^3 > f(1 - 5x).

Simplifying further involves expanding and collecting like terms. For instance, \[f(1 - 5x) = 1 - (1 - 5x) - (1 - 5x)^3\] simplifies to \[5x - 1 + 15x - 75x^2 + 125x^3\]. Similarly, \[1 - (1 - x - x^3)^3\] needs individual term expansion and simplification. The simplified version of the polynomials makes solving the inequality more feasible. Key steps:

• Expand all parentheses and combine like terms
• Use polynomial identities if applicable
• Reduce the expressions by combining terms

This will help in achieving a simpler polynomial which is crucial for interval testing.

interval testing

Interval testing involves evaluating the inequality within specific ranges of the variable to see where the inequality holds true. After simplifying our inequality: \[x + x^3 - (1 - x - x^3)^3 > 5x - 1 + 15x - 75x^2 + 125x^3\], we must now determine which intervals fulfill this condition.

Testing intervals:

• Choose intervals based on the given options
• Substitute values from these intervals into the simplified inequality
• Check if the inequality holds true within these intervals

In our problem, testing the intervals shows that the inequality is satisfied in the interval \((0,2)\).

Remember, where needed, also check the endpoints of these intervals. This systematic approach helps in finding the solution set of the inequality.