Problem 188 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ x(x-1)^{2}>0 $$

Step-by-Step Solution

Answer

The solution to the inequality \(x(x-1)^{2} > 0\) is \(x \in (-\infty, 0) \cup (1, \infty)\).

Step 1: Identify the critical points (roots)

To find the critical points, set the equation equal to zero: \( x(x-1)^{2}=0 \) We have two factors here, x and \((x-1)^{2}\). The roots occur when either of them is equal to zero. So, the roots are x = 0 and x = 1.

Step 2: Test the intervals between the roots

Now, we'll test the intervals between and around these critical points to see which intervals satisfy the inequality: 1. Interval \((-∞, 0)\) Choose a test point, say x = -1: \((-1)((-1)-1)^{2} > 0\) \(1 * 4 > 0\) which is true. 2. Interval \((0, 1)\) Choose a test point, say x = 0.5: \((0.5)((0.5)-1)^{2} > 0\) \(0.5 * 0.25 > 0\) which is false. 3. Interval \((1, ∞)\) Choose a test point, say x = 2: \((2)((2)-1)^{2} > 0\) \(2 * 1 > 0\) which is true.

Step 3: Write down the solution

From our test results, the inequality \(x(x-1)^{2} > 0\) holds true for the intervals \((-∞, 0)\) and \((1, ∞)\). Therefore, the solution is: \[x \in (-\infty, 0) \cup (1, \infty)\].

Key Concepts

Critical Points

To solve the inequality \(x(x-1)^2 > 0\), the first step is to find the critical points or roots of the equation. These are the values of \(x\) where the expression equals zero. By setting the inequality to zero, we have:\[x(x-1)^2 = 0\]The expression can be factored into two distinct factors: \(x\) and \((x-1)^2\). A product of multiple factors equals zero only when at least one of the factors is zero. So, we solve each factor separately:

• \(x = 0\), because \(x=0\) causes the entire product to be zero.
• \((x-1)^2 = 0\) gives \(x=1\). The square \((x-1)^2\) also causes the product to be zero when \(x = 1\).

Thus, the critical points for this inequality are \(x = 0\) and \(x = 1\). These are the points where the behavior of the inequality might change.

Interval Testing

After finding the critical points, the next step is interval testing. This involves testing the sign of the inequality in the intervals created by these critical points. The critical points \(x=0\) and \(x=1\) divide the number line into three intervals:

• \((-∞, 0)\)
• \((0, 1)\)
• \((1, ∞)\)

For each interval, choose a test point and substitute it into the inequality \(x(x-1)^2 > 0\):- **Interval \((-∞, 0)\):** Let's use the test point \(x = -1\). Calculating: \[(-1)((-1)-1)^2 = 1 * 4 > 0\], which returns true. - **Interval \((0, 1)\):** Let's try \(x = 0.5\). Evaluating: \[0.5((0.5)-1)^2 = 0.5 * 0.25 > 0\], which results in false. - **Interval \((1, ∞)\):** Use \(x = 2\). Testing gives: \[2((2)-1)^2 = 2 * 1 > 0\], which is true. This testing process helps determine where the inequality holds true.

Inequality Solution

The solution to the inequality \(x(x-1)^2 > 0\) involves compiling the intervals where the inequality is satisfied. From the interval testing:- In the interval \((-∞, 0)\), the inequality returned true, meaning it satisfies the condition \( > 0\).- In the interval \((0, 1)\), the inequality returned false, which means this interval does not satisfy the inequality.- In the interval \((1, ∞)\), the inequality tested true, again satisfying the condition \( > 0\).Therefore, the solution to the inequality, in interval notation, is:\[x \, \in \, (-Infty, 0) \cup (1, Infty)\]This notation compactly represents the union of intervals where the inequality holds true, excluding the critical points themselves.

Roots Identification

Understanding roots identification helps in recognizing intervals where the inequality changes sign. In the equation \(x(x-1)^2 = 0\), identifying the roots is pivotal because these are the points at which the expression is either zero or changing sign.

• The root \(x = 0\) comes from solving \(x = 0\).
• The root \(x = 1\) arises from solving \((x-1)^2 = 0\).

Each root divides the number line into separate intervals, forming sections that need individual testing. Understanding these roots is essential since they represent the boundary or transition points. This means the function switches from positive to negative or vice versa at these points, hence why the inequality might not hold exactly at the root values. These identified roots, therefore, play a crucial role in determining the overall solution to the inequality \(x(x-1)^2 > 0\).