Problem 56 - Calculus with Concepts in Calculus

Question

Show that the given inequality holds for the given values of \(\mathrm{x}\). $$ 4 x^{2}+1 / x>5 \text { for } x>1 $$

Step-by-Step Solution

Answer

The inequality holds for all \(x > 1\) since \(4x^3\) dominates the sum.

Step 1: Understand the Inequality

We need to show that the inequality \(4x^2 + \frac{1}{x} > 5\) holds true for values of \(x > 1\). This means substituting any number greater than 1 into the expression should result in a value greater than 5.

Step 2: Substitute Test Values

Let's substitute \(x = 2\) as a test case since \(x = 2 > 1\). For \(x = 2\), the expression becomes:\[4(2)^2 + \frac{1}{2} = 16 + 0.5 = 16.5\]Since 16.5 is greater than 5, the inequality holds for \(x = 2\).

Step 3: Analyze the Expression

The expression \(4x^2 + \frac{1}{x}\) consists of two parts: \(4x^2\) and \(\frac{1}{x}\). As \(x\) increases, \(4x^2\) increases quadratically, and \(\frac{1}{x}\) decreases, making the positive contribution of \(4x^2\) dominant. This suggests the expression will be greater than 5 for \(x > 1\).

Step 4: Solve Algebraically

To solve algebraically, consider shifting all terms to one side: \[4x^2 + \frac{1}{x} - 5 > 0\]Multiply through by \(x\) (valid for \(x > 1\) and thus \(x > 0\)) to eliminate the fraction:\[4x^3 - 5x + 1 > 0\]We know \(4x^3\) grows much faster than \(-5x\) and \(1\) will have less impact as \(x\) increases from 1, making the inequality true.

Step 5: Verify That It Holds for All x > 1

For larger values of \(x\), since \(4x^3\) increases more rapidly than \(-5x\) can negatively impact the sum, the inequality \(4x^3 - 5x + 1 > 0\) will certainly hold for every \(x > 1\). A small numerical check for values like \(x = 1.5\) also supports this, as the result is greater than 5.

Key Concepts

quadratic inequalities

Quadratic inequalities are those in which the variable is raised to the second power, leading to expressions like \(4x^2\). Understanding these inequalities is essential because they help describe areas where the function is either above or below a certain line, such as \(y = 5\) in our example. When working with these inequalities, you often have to analyze the expression to determine the range of values that satisfy the inequality.

• Quadratic terms influence the shape of the graph, typically forming a parabolic curve.
• Solving these inequalities usually involves finding intervals where the function is positive or negative.

In the exercise provided, we compare the expression \(4x^2 + \frac{1}{x}\) to the constant 5, focusing especially on values of \(x > 1\). Here, the quadratic term \(4x^2\) ensures the function grows rather quickly as \(x\) increases.

test values in inequality

Testing values in an inequality is a practical method to confirm if an inequality holds true for certain points within the range of interest. In this process, you plug specific numbers into the inequality to see whether they satisfy the condition given.

• Choose values that are simple and easy to calculate, like integers or common fractions.
• This often provides a quick confirmation before more complex algebraic proofs.

For instance, in proving our inequality \(4x^2 + \frac{1}{x} > 5\), we substituted \(x = 2\), resulting in a value of 16.5. This demonstrates that the inequality holds for \(x = 2\), a suitable strategy when dealing with continuous ranges of values where a quick check is needed.

analyzing algebraic expressions

Analyzing algebraic expressions helps us understand the behavior of different components within those expressions. Here, we need to isolate how each part—such as quadratic or rational terms—affects the overall expression.

• Focus on growth rates of different terms as \(x\) increases or decreases.
• Understand which terms dominate as the variable moves across specified ranges.

In the expression \(4x^2 + \frac{1}{x}\), observe that \(4x^2\) increases rapidly due to its quadratic nature, while \(\frac{1}{x}\) decreases as \(x\) becomes larger. This shift in growth rates implies that the quadratic term will outrun the decreasing fraction, making the whole expression grow beyond 5 for \(x > 1\). This balance is crucial in evaluating inequalities effectively.

inequality proof techniques

Inequality proof techniques involve demonstrating that an inequality holds for a given set of values through algebraic manipulation or logical reasoning. These techniques are essential in verifying statements and ensuring their validity under all circumstances.

• Bring all terms to one side of the inequality to simplify analysis.
• Eliminate fractions when possible, especially for domains where the variable is positive.

In solving \(4x^2 + \frac{1}{x} > 5\), transforming it into another inequality, such as \(4x^3 - 5x + 1 > 0\), helps clarify its truth. Here, observing how \(4x^3\) grows much faster than \(-5x\) and \(+1\) provides proof that the expression increases for all \(x > 1\). Algebraic transformations aid in understanding how different factors interact within inequalities, confirming the conditions under which they hold true.