# 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:

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

- (a) is a critical point of f(x) and
- (c) f(a) is a local minimum

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#### Learn more about this topic:

from Math 104: Calculus

Chapter 9 / Lesson 4