use the intermediate value theorem to identify an interval

Question

use the intermediate value theorem to identify an interval where the equation sinx- x^2= 0 has a non-zero solution

Answer 1

An equation has a non zero solution if the values do not change signs or does not become zero.  We know that sine is positive in 1st and 2nd quadrants.  Negative in the 3rd and 4th quadrants.

 

Lets pick a reference angle, such as x=π/4.  We will evaluate the equation at the angle at all 4 quadrants.

 

sinx - (x²) = 0

 

We know that x² always positive, and that x is a value less than 1.

 

At x=0 , the result is zero

At x=π/3 , the result is negative.

At x=π/4, the result is positive.

At x=π/6 , the result is positive.

At x=π/2 , the result is negative.

 

Between x=π/4 and x=π/6, the values do not change signs.  This is one interval where the solution is non-zero.