Question

Solve the inequality and write the solution set in interval notation. $4x^3-x^4\ge 0$

Short answer

The solution to the inequality $4x^3-x^4\ge 0$ in interval notation is (0, 4] ∪ [4, ∞).

Step-by-step solution

Factorise the Polynomial

The first step is to factorise the given polynomial. In this inequality, it can be factorised by factoring out the greatest common factor, $x^3$, from each term. This results in $x^3(4-x)\ge 0$

Determine Critical Points

Critical points are solutions when the inequality equals 0. By setting $x^3$ and $4-x$ to zero, the critical values are 0 and 4.

Test Intervals

Create intervals using the critical numbers which will then be tested in the inequality. The intervals are (-∞, 0), (0, 4), (4, ∞). Pick a test number from each interval and substitute into the inequality. For (-∞, 0), a value of -1 can be chosen, for (0, 4), a value of 1, and for (4, ∞), a value of 5. After testing these values, the inequality holds true for intervals (0, 4) and (4, ∞) but does not hold true for interval (-∞, 0)

Write Answer in Interval Notation

After obtaining the intervals where the inequality holds true, write the answers in interval notation as a final answer. For the given inequality, the solution set will be in the form (0, 4] ∪ [4, ∞).