Q. 87 Use the Intermediate Value Theor... [FREE SOLUTION]

Chapter 10: Q. 87 (page 777)

Use the Intermediate Value Theorem to prove that every cubic function $f (x) = A x^{3} + B x^{2} + C x + D$ has at least one real root. You will have to first argue that you can find real numbers a and b so that f(a) is negative and f(b) is positive.

Short Answer

Expert verified

In either case, the intermediate value theorem applies to the continuous cubic function $f (x)$ on $[- N, N]$ , and therefore $f (x)$ has at least one real root on $[- N, N)$ .

Step by step solution

Step 1. Given Information.

Given expression $f (x) = A x^{3} + B x^{2} + C x + D$ $A x^{3} + B x^{2} + C x + D$

Step 2. The strategy is to prove that every cubic function stated above has at least one real root.

If $A > 0$ then for large magnitude N we will have $f (- N) < 0 < f (N)$

And if $A < 0$ then for large magnitude $N$ we will have $f (- N) < 0 < f (N)$

In either case, the intermediate Value theorem applies to the continuous cubic function $f (x)$ on $[- N, N]$ and therefore $f (x)$ has at least one real root on $[- N, N]$

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91影视

Use the Intermediate Value Theorem to prove that every cubic function $f (x) = A x^{3} + B x^{2} + C x + D$ has at least one real root. You will have to first argue that you can find real numbers a and b so that f(a) is negative and f(b) is positive.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. The strategy is to prove that every cubic function stated above has at least one real root.

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91影视

Use the Intermediate Value Theorem to prove that every cubic function f(x)=Ax3+Bx2+Cx+Dhas at least one real root. You will have to first argue that you can find real numbers a and b so that f(a) is negative and f(b) is positive.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. The strategy is to prove that every cubic function stated above has at least one real root.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Applied Mathematics

Geometry

Theoretical and Mathematical Physics

Statistics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Use the Intermediate Value Theorem to prove that every cubic function $f (x) = A x^{3} + B x^{2} + C x + D$ has at least one real root. You will have to first argue that you can find real numbers a and b so that f(a) is negative and f(b) is positive.