Power functions and first sketches practice question 3
The curves and meet at three points. Find the largest -coordinate at which they meet.
Two curves meet where they give the same for the same . How can you find all three such without losing one along the way?
Set the two formulas equal and move everything to one side. Do not divide by : you would lose a solution.
A whole number.
Theory UBC Math 100 §1.1Key results
Worked solution
Strategy
Two curves meet where their formulas agree. Set them equal, factor, and read off every solution; the question wants the largest.
Working
Bring both terms to one side so the equation can be factored.
Take out the common factor 2x, then factor the difference of squares.
Three meeting points, as promised. The largest is x = 2.
Answer
The answer is 2.
Check
At x = 2 both curves are at height 16, and for x > 2 the cubic stays above 8x since 2x(x² − 4) > 0, so they never meet again.
Where this goes wrong
Dividing both sides by x at the start. That discards the solution x = 0, and although this question only asks for the largest, a question asking for all three would lose a mark.
The idea behind it
The power functions and first sketches lesson, a figure to move, in a few steps
Try a similar question
Find the positive value of at which the curves below intersect.Saved in this browser only.