Use the intermediate value theorem to determine whether the function

skomminbv

skomminbv

Answered question

2021-11-30

Use the intermediate value theorem to determine whether the function f(x)=x3+2x4 has a root or not. between x=1 and x=2. Ifyes, then find the root to five decimal places.

Answer & Explanation

Muspee

Muspee

Beginner2021-12-01Added 13 answers

Step1
Given,
f(x)=x3+2x4 over interval, [1, 2]
Definition of Intermediate value theorem:
If f(x) be the continuous function on the closed interval [a,b]
then, f(a) and f(b) has opposite sides in the interval,
then there must be some value x=c lies on the interval [a,b] for which f(c)=0
Step2
Plug the endpoints of intervals in the given function:
Plug x=1 we get,
f(1)=13+2(1)4
=1+24
=34
=1
Plug x=2 we get,
f(2)=23+2(2)4
=8+44
=8
From the above definition of intermediate value theorem we get,
f(1)<0,f(2)>O then f(x) is continuous on the closed interval [1,2],
there must be some value x=c lies on the interval [1,2] for which f(c)=0
Step3
Put x=c on the given function and equate to zero we get,
f(c)=c3+2c4=0
The below graph is the graph of the function,
From the graph, we get at f(x}=0 we get a value of x=1.17951
Therefore roots of the given function are,
x=1.17951
The given equation is the equation of polynomial with degree 3 so it has 3 roots.
Now we get one of the roots of the polynomial from the graph we clearly see that the other two graphs are imaginary.
Therefore the roots are x=1.17951
image

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