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Function analysis using derivatives
This test uses the first derivative to analyse a function. Tasks include finding critical points, intervals of increase and decrease, local extrema and absolute maximum or minimum values.
Questions: 10Estimated time: 17 minutes
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Question 1out of 10
Which sequence is appropriate when analysing a function with derivatives?
Find the domain and critical points, then the derivative signs on intervals. f′ > 0 means increasing and f′ < 0 decreasing. A change from − to + gives a minimum; + to − a maximum. f′ = 0 alone does not guarantee an extremum. For f′ = 2(x − r), the minimum is at r.
Question 2out of 10
On [−2, 6], f(x) = (x − 1)². Find the maximum value.
On a closed interval, evaluate the function at both endpoints and all critical points inside it. Compare function values, not x values. For positive m, m(x − r)² is greatest at the endpoint farther from r.
Question 3out of 10
What extremum occurs at the vertex of f(x) = 1x²?
Find the domain and critical points, then the derivative signs on intervals. f′ > 0 means increasing and f′ < 0 decreasing. A change from − to + gives a minimum; + to − a maximum. f′ = 0 alone does not guarantee an extremum. For f′ = 2(x − r), the minimum is at r.
Question 4out of 10
For f(x) = 2x² − 16x, find the critical x where f′(x) = 0.
Find the domain and critical points, then the derivative signs on intervals. f′ > 0 means increasing and f′ < 0 decreasing. A change from − to + gives a minimum; + to − a maximum. f′ = 0 alone does not guarantee an extremum. For f′ = 2(x − r), the minimum is at r.
Question 5out of 10
How is the derivative sign related to increasing and decreasing?
Find the domain and critical points, then the derivative signs on intervals. f′ > 0 means increasing and f′ < 0 decreasing. A change from − to + gives a minimum; + to − a maximum. f′ = 0 alone does not guarantee an extremum. For f′ = 2(x − r), the minimum is at r.
Question 6out of 10
What extremum occurs at the vertex of f(x) = −1x²?
Find the domain and critical points, then the derivative signs on intervals. f′ > 0 means increasing and f′ < 0 decreasing. A change from − to + gives a minimum; + to − a maximum. f′ = 0 alone does not guarantee an extremum. For f′ = 2(x − r), the minimum is at r.
Question 7out of 10
The derivative is f′(x) = 2(x − 1). Where is f increasing?
Find the domain and critical points, then the derivative signs on intervals. f′ > 0 means increasing and f′ < 0 decreasing. A change from − to + gives a minimum; + to − a maximum. f′ = 0 alone does not guarantee an extremum. For f′ = 2(x − r), the minimum is at r.
Question 8out of 10
For f(x) = 5x² + 60x, find the critical x where f′(x) = 0.
Find the domain and critical points, then the derivative signs on intervals. f′ > 0 means increasing and f′ < 0 decreasing. A change from − to + gives a minimum; + to − a maximum. f′ = 0 alone does not guarantee an extremum. For f′ = 2(x − r), the minimum is at r.
Question 9out of 10
The derivative is f′(x) = 2(x + 3). Where is f decreasing?
Find the domain and critical points, then the derivative signs on intervals. f′ > 0 means increasing and f′ < 0 decreasing. A change from − to + gives a minimum; + to − a maximum. f′ = 0 alone does not guarantee an extremum. For f′ = 2(x − r), the minimum is at r.
Question 10out of 10
Is every point where f′(x) = 0 necessarily an extremum?
Find the domain and critical points, then the derivative signs on intervals. f′ > 0 means increasing and f′ < 0 decreasing. A change from − to + gives a minimum; + to − a maximum. f′ = 0 alone does not guarantee an extremum. For f′ = 2(x − r), the minimum is at r.
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Sample questions
- For f(x) = ___x² + (___)x, find the critical x where f′(x) = 0.
- The derivative is f′(x) = 2(x − (___)). Where is f increasing?
- The derivative is f′(x) = 2(x − (___)). Where is f decreasing?
- What extremum occurs at the vertex of f(x) = ___x²?
- On [___, ___], f(x) = ___(x − (___))². Find the maximum value.
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