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Given that f(x) is continuous everywhere and changes from negative to positive at x = a, which of...

Question:

Given that f(x) is continuous everywhere and changes from negative to positive at x = a, which of the following statements must be true?

(a) a is a critical point of f(x)

(b) f(a) is a local maximum

(c) f(a) is a local minimum

(d) f (a) is a local maximum

(e) f (a) is a local minimum

Relative Extremum:

  1. A critical point is a point where the first derivative of the function is zero.
  2. At this point, the function has either a local minimum or a local maximum.
  3. If the derivative changes its sign from positive to negative, the function has a local maximum.
  4. If the sign of the derivative changes from negative to positive at the critical point, the function has a local minimum at that point.

Answer and Explanation:

In the given statement, it has been said that the function changes from negative to positive. It did not say about the first derivative of the function.

So, if a function changes from negative to positive, it does not tells us anything about the minimum, or maximum or critical point.

However, if we change the statement to Given that f(x) is continuous everywhere and its first derivative changes from negative to positive at x = a,

Then there is an accurate conclusion of this statement.

  1. (a) is a critical point of f(x) and
  2. (c) f(a) is a local minimum

Learn more about this topic:

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Using Differentiation to Find Maximum and Minimum Values

from Math 104: Calculus

Chapter 9 / Lesson 4
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