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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Show that the equation x4 + 4x + c = 0 has at most two real roots.

To determine

To show: The given equation has at most two real roots.

Explanation

Given:

The equation is x4+4x+c=0 .

Theorem used: Rolle’s Theorem

“If a function f satisfies the following conditions,

1. A function f is continuous on the closed interval [a,b] .

2. A function f is differentiable on the open interval (a,b) .

3. f(a)=f(b)

Then, there is a number c in open interval (a,b) such that f(c)=0 ”.

Proof:

Suppose f(x) has three distinct real roots a, b and r where a<b<r . Then f(a)=f(b)=f(r)=0 .

Since the polynomial is continuous on the closed interval [a,b] and [b,r] and differentiable on the open interval (a,b) and (b,r) , then by Roll’s Theorem, there exist a number c1 and c2 with a<c1<b and a<c2

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