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Flip to reveal answersDefine a local maximum of a function.
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All 16 Flashcards — Local maxima and minima
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Question
Define a local maximum of a function.
Answer
A point where the function value is higher than all nearby values — the graph has a peak there. The function increases up to it and decreases after it.
Question
What is the difference between a maximum point and a maximum value?
Answer
Maximum point: both coordinates, e.g. (2, 9). Maximum value: just the y-value, e.g. 9. IB questions ask for either — read carefully.
Question
At a turning point, what is true about the gradient of the curve?
Answer
The gradient is zero at every turning point. The tangent line is horizontal there.
Question
Can a function have a local maximum that is lower than a local minimum elsewhere on the curve?
Answer
Yes — local max/min are only local (in a neighbourhood). The global maximum is the highest point overall, which may be different from any local maximum.
Question
The graph has a peak at (3, 8). Write down the local maximum.
Answer
Local maximum at (3, 8). The x-coordinate is 3 and the maximum value is 8. State both.
Question
IB asks "Write down the coordinates of the local minimum." What must your answer look like?
Answer
A coordinate pair: e.g. (−1, −5). Both x and y must be stated. Writing only x = −1 loses the second mark.
Question
A graph reaches a low point at (−2, 1). What is the minimum value of f?
Answer
1. The minimum value is the y-coordinate. The point (−2, 1) tells you the minimum occurs at x = −2, and the minimum value is 1.
Question
How do you identify a local minimum from a graph just by looking?
Answer
Look for a trough — a point where the graph changes from decreasing (falling) to increasing (rising). The curve dips down then comes back up.
Question
Steps to find a local maximum on a GDC (TI-84):
Answer
1. Graph f(x) with an appropriate window. 2. Press 2nd → Calc → Maximum. 3. Move left of the peak: press Enter (left bound). 4. Move right of the peak: press Enter (right bound). 5. Press Enter again (guess). GDC shows coordinates.
Question
GDC gives a minimum at x = 2.718. IB asks for the answer to 3 s.f. What do you write?
Answer
x = 2.72 (3 s.f.). Then substitute into f to find y, e.g. y = f(2.72). State both coordinates.
Question
Why must you always state y as well as x for a turning point?
Answer
IB markschemes award separate marks for each coordinate. Writing only x earns 0 marks for the y-coordinate. Always give both.
Question
A cubic has two turning points. GDC Maximum gives (−1, 4). What else should you find?
Answer
The local minimum. Run GDC Minimum with bounds around the other turning point to get its coordinates too.
Question
h(t) = −4t² + 24t. The maximum is at (3, 36). Interpret this in context.
Answer
After 3 seconds the ball reaches its highest point of 36 m above the ground.
Question
Profit P(n) has a maximum at (500, 8000). What does this mean?
Answer
Maximum profit of 8000 occurs when 500 units are produced. Producing more or fewer reduces profit.
Question
IB asks "Interpret the local maximum in context." How do you score the mark?
Answer
State what the x-value represents (e.g. time, units) and what the y-value represents (e.g. height, profit) using the context's specific units. E.g. "After 3 hours, temperature reaches its peak of 36°C."
Question
A profit model has a minimum at n = 10. What does this suggest about the business?
Answer
At n = 10 units, profit is at its lowest. The business loses the most money at this production level, and should either produce fewer or more units.
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