Use intermediate value theorem and interval bisection method to prove the following equation has at least 3 solutions

Question

I'm just learnt about the Intermediate Value Theorem and I'm solving this question:

Explain why the equation 2x - 3x⁵ = 1/4 has at least three solutions within the interval [-1, 1] using Intermediate Value Theorem and Interval Bisection Method.

Thanks a lot for your help!

Answer 1

2x - 3x^5 = 1/4

8x - 12x^5 = 1

12x^5 - 8x + 1 = 0

So f(x) = 12x^5 - 8x + 1 is the CONTINUOUS polynomial

function whose solutions and zeros are of interest.

There's one sign change so there is at most one solution

The approximate function values show:

f(-1) = -3 and f(-0.9) = 1.11412

so IVT guarantees a solution on (-1 , -0.9)

f(0.1) = 0.20012 and f(0.2) = -0.59616

so IVT guarantees a solution on (0.1,0.2)

f(0.8) = -0.88588 and f(0.9) = 5

so IVT guarantees a solution on (0.8,0.9)

Examining the graphs in Desmos, the approximate

solutions are x = -0.9325, x = 0.12505, and x = 0.8692

Bisection method shows the solutions as

x=-0.932471279863982441

x=0.125045860401730025

x=0.869199905146805565