Practice Flashcards
What does it mean for a function to be INCREASING on an interval?
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All Flashcards in Topic 5.2
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5.2.18 cards
What does it mean for a function to be INCREASING on an interval?
f is increasing if f'(x) > 0 for all x in that interval. As x gets bigger, f(x) gets bigger — the graph goes UP.
Positive derivative = going up.
What does it mean for a function to be DECREASING on an interval?
f is decreasing if f'(x) < 0 for all x in that interval. As x gets bigger, f(x) gets smaller — the graph goes DOWN.
Negative derivative = going down.
How do you find where a function is increasing or decreasing?
1) Find f'(x). 2) Solve f'(x) = 0 — these are the critical x-values. 3) Test a value in each interval: if f'(x) > 0, increasing; if f'(x) < 0, decreasing.
Critical points divide the number line into intervals.
f(x) = x² − 4x. Where is it increasing? Where is it decreasing?
f'(x) = 2x − 4. Critical point: x = 2. For x < 2: f'(x) < 0 → DECREASING. For x > 2: f'(x) > 0 → INCREASING.
Solve f'(x)=0, then test each side.
What does f'(x) = 0 tell you about increasing/decreasing?
It marks the boundary between increasing and decreasing. At that point, the function is momentarily flat — it is a critical (stationary) point.
f'(x)=0 is the turning-point signal.
What is the sign diagram method?
Draw a number line. Mark critical x-values. Pick one test x in each interval, evaluate f'(x). Label each interval + (increasing) or − (decreasing).
One test point per interval is enough.
f(x) = −x² + 6x. Is f increasing at x = 2?
f'(x) = −2x + 6. Substitute x = 2: f'(2) = 2 > 0. Yes — f is increasing at x = 2.
Substitute x into f'(x) and check the sign.
If f'(x) > 0 everywhere, what does that mean for the function?
The function is increasing for all x. It never turns around. Example: f(x) = x³ has f'(x) = 3x² ≥ 0 but is still overall increasing.
Always increasing = positive derivative throughout.
Topic 5.2 study notes
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