Math  /  Calculus

QuestionWhich of the following are true? A) It is possible for a function to not have a global maximum or a global minimum. B) If a continuous function has 3 critical points then it must have a global maximum. Only B Only A AA and BB Neither A or B Clear my selection

Studdy Solution

STEP 1

1. We are considering properties of functions, particularly with respect to global maxima and minima.
2. A function may or may not be continuous.
3. Critical points are where the derivative is zero or undefined.

STEP 2

1. Evaluate statement A.
2. Evaluate statement B.
3. Determine which statements are true.

STEP 3

Evaluate statement A: "It is possible for a function to not have a global maximum or a global minimum."
Consider a function that is not bounded, such as f(x)=x f(x) = x . This function increases without bound as x x \to \infty and decreases without bound as x x \to -\infty . Therefore, it does not have a global maximum or a global minimum.

STEP 4

Evaluate statement B: "If a continuous function has 3 critical points then it must have a global maximum."
A continuous function with 3 critical points does not necessarily have a global maximum. For example, consider a polynomial function that has critical points but is unbounded, such as f(x)=x33x f(x) = x^3 - 3x . This function has critical points but no global maximum.

STEP 5

Determine which statements are true:
Statement A is true because there exist functions without global maxima or minima, such as f(x)=x f(x) = x .
Statement B is false because a continuous function with 3 critical points does not guarantee a global maximum, as demonstrated by f(x)=x33x f(x) = x^3 - 3x .
Thus, the correct answer is "Only A".
The correct answer is:
Only A \boxed{\text{Only A}}

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