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Questions and Answers
What is a critical point of a function f of a variable x?
What is a critical point of a function f of a variable x?
The x coordinate of a relative min or max
A continuous function on a closed interval can have only one maximum value.
A continuous function on a closed interval can have only one maximum value.
True (A)
If the second derivative of a function is always positive, the function must have a relative minimum value.
If the second derivative of a function is always positive, the function must have a relative minimum value.
False (B)
If a function f has a relative minimum at x=c, then f'(c) is zero.
If a function f has a relative minimum at x=c, then f'(c) is zero.
If f'(2)=0 and f''(2)>0 for all x in the interval, then the absolute maximum will occur at the right endpoint of the interval.
If f'(2)=0 and f''(2)>0 for all x in the interval, then the absolute maximum will occur at the right endpoint of the interval.
The absolute minimum of a continuous function on a closed interval can occur at only one point.
The absolute minimum of a continuous function on a closed interval can occur at only one point.
If x=2 is the only critical point of a function f and f''(2)>0, then f(2) is the minimum value of the function.
If x=2 is the only critical point of a function f and f''(2)>0, then f(2) is the minimum value of the function.
To locate the absolute extrema of a continuous function on a closed interval, you need only compare the y values of all critical points.
To locate the absolute extrema of a continuous function on a closed interval, you need only compare the y values of all critical points.
If f'(c)=0 and f'(x) decreases through x=c, then x=c locates a relative minimum value of the function.
If f'(c)=0 and f'(x) decreases through x=c, then x=c locates a relative minimum value of the function.
Absolute extrema of a continuous function on closed intervals can occur at only endpoints or critical points.
Absolute extrema of a continuous function on closed intervals can occur at only endpoints or critical points.
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Study Notes
Critical Points Definition
- A critical point of function ( f ) is an ( x ) coordinate where a relative minimum or maximum occurs.
Maximum Values in Continuous Functions
- A continuous function on a closed interval can have only one maximum value, though multiple ( x ) values can correspond to it.
Behavior of Derivatives
- If the second derivative ( f'' ) is always positive, ( f ) has a relative minimum, contradicting the idea that only the first derivative ( f' ) must be zero.
Critical Points and Derivative Values
- A function having a relative minimum at ( x=c ) implies that the derivative ( f'(c) ) equals zero.
Conditions for Absolute Maximum
- If ( f'(2)=0 ) and ( f''(2)>0 ), the absolute maximum occurs at the right endpoint of the interval.
Absolute Minimum Uniqueness
- The absolute minimum of a continuous function on a closed interval can occur at multiple points, contrary to the notion that it must be at only one point.
Critical Points and Minimum Values
- If ( x=2 ) is the sole critical point and ( f''(2)>0 ), then ( f(2) ) represents the minimum value of the function.
Finding Absolute Extrema
- To determine absolute extrema of a continuous function on a closed interval, it is necessary to evaluate all critical points and endpoints, not just the critical points alone.
Relative Maximum and Decreasing Derivatives
- When ( f'(c)=0 ) and ( f'(x) ) decreases through ( x=c ), this indicates that ( x=c ) locates a relative maximum.
Absolute Extrema Locations
- Absolute extrema of continuous functions on closed intervals can occur at endpoints or critical points, highlighting the importance of both in calculations.
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