Questions and Answers
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.
True
If the second derivative of a function is always positive, the function must have a relative minimum value.
False
If a function f has a relative minimum at x=c, then f'(c) is zero.
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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.
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The absolute minimum of a continuous function on a closed interval can occur at only one point.
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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.
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To locate the absolute extrema of a continuous function on a closed interval, you need only compare the y values of all critical points.
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If f'(c)=0 and f'(x) decreases through x=c, then x=c locates a relative minimum value of the function.
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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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Description
This quiz covers critical points, maximum values, and the behavior of derivatives in calculus. Explore key concepts such as relative minima, maxima, and the conditions for absolute extremes in continuous functions. Test your understanding of how derivatives relate to the characteristics of a function.
