SOLUTION: Please help me solve this question: Show that the equation has at least one real root. a) cosx=x^3 b)x^(5)-x^(2)+2x+3=0

Algebra ->  Functions -> SOLUTION: Please help me solve this question: Show that the equation has at least one real root. a) cosx=x^3 b)x^(5)-x^(2)+2x+3=0      Log On


   

Question 734675: Please help me solve this question:
Show that the equation has at least one real root.
a) cosx=x^3
b)x^(5)-x^(2)+2x+3=0
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
a) Consider f%28x%29=cos%28x%29-x%5E3
It is a continuous function.
f%280%29=cos%280%29-0%5E3=1%3E0
f%28pi%2F2%29=cos%28pi%2F2%29-%28pi%2F2%29%5E3=0-pi%5E3%2F8=-pi%5E3%2F8%3C0
The continuous function f%28x%29 is positive for x=0 and negative for x=pi%2F2 so it must have a zero somewhere in the (0,pi%2F2) interval (according to the intermediate value theorem).

You could do something similar with g%28x%29=x%5E5-x%5E2%2B2x%2B3%3E0.
You can easily see that g%280%29=3%3E0 and g%28-10%29=-100000%2B100%2B20%2B3%3C0 and that is as much as is needed to prove is that there is one real root.