Questions: The function h is given by h(x)=x^4-6x^2 and the function k is given by k(x)=3x^2. Find all intervals where h(x)>k(x).

Transcript text: 20. The function $h$ is given by $h(x)=x^{4}-6 x^{2}$ and the function $k$ is given by $k(x)=3 x^{2}$. Find all intervals where $h(x)>k(x)$.

Solution

Solution Steps

Step 1: Set Up the Inequality

Given the functions $ h(x) = x^4 - 6x^2 $ and $ k(x) = 3x^2 $, we need to find the intervals where $ h(x) > k(x) $. This translates to solving the inequality: \[ x^4 - 6x^2 > 3x^2 \]

Step 2: Simplify the Inequality

Simplify the inequality by combining like terms: \[ x^4 - 9x^2 > 0 \]

Step 3: Factor the Inequality

Factor the expression on the left-hand side: \[ x^2(x^2 - 9) > 0 \] Further factorize: \[ x^2(x - 3)(x + 3) > 0 \]

Step 4: Find the Critical Points

The critical points are the roots of the equation $ x^2(x - 3)(x + 3) = 0 $: \[ x = 0, \, x = 3, \, x = -3 \]

Step 5: Test Intervals Between the Critical Points

Test the intervals determined by the critical points: $ (-\infty, -3) $, $ (-3, 0) $, $ (0, 3) $, and $ (3, \infty) $.