Math Problem Statement

f is a continuous function such that f(1) = 3 and f(7) = - 2 then its graph must intersect the x-axis. Give counter argument

Solution

The assertion is based on the Intermediate Value Theorem (IVT), which states that if a function ff is continuous on a closed interval [a,b][a, b], and if f(a)f(a) and f(b)f(b) have opposite signs (i.e., one is positive and the other is negative), then there must exist at least one point c(a,b)c \in (a, b) where f(c)=0f(c) = 0.

In this case:

  • f(1)=3f(1) = 3 (positive),
  • f(7)=2f(7) = -2 (negative).

Given these values, the IVT would suggest that there must be some c(1,7)c \in (1, 7) where f(c)=0f(c) = 0, meaning the graph intersects the x-axis at least once in the interval (1,7)(1, 7).

Counter Argument

A potential counterargument would involve the function not satisfying the hypothesis of the IVT, specifically discontinuity. Even though the problem states that ff is continuous, for the sake of argument, we could assume that the function might exhibit some form of discontinuity in the interval (1,7)(1, 7). For example, if the function had a jump discontinuity between x=1x = 1 and x=7x = 7, it might change from 3 to -2 without ever crossing zero, although this contradicts the given condition of ff being continuous.

Since the problem guarantees continuity, it cannot be disproven under normal conditions, and any counterargument would rely on assuming the function is not continuous, which violates the premise of the question.

Would you like more details on this, or any other clarification?

Related Questions:

  1. Can the graph of a continuous function cross the x-axis more than once?
  2. How does the IVT apply to non-linear continuous functions?
  3. What happens if f(1)=f(7)f(1) = f(7) but both are nonzero?
  4. Can you describe a function that touches the x-axis but does not cross it?
  5. How does continuity affect the range of a function on a closed interval?

Tip:

When applying the Intermediate Value Theorem, always ensure that the function is continuous on the interval and that the function values at the endpoints have opposite signs.

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Math Problem Analysis

Mathematical Concepts

Calculus
Continuous Functions
Intermediate Value Theorem (IVT)

Formulas

-

Theorems

Intermediate Value Theorem

Suitable Grade Level

Grades 11-12, Early College