Use the graph of the derivative f ′ of a continuous function f is shown. (Assume f ′ continues to ∞.) (b) At what value(s) of x does f have a local maximum? (Enter your answers as a comma-separated list.) At what value(s) of x does f have a local minimum? (Enter your answers as a comma-separated list
Use the graph of the derivative f ′ of a continuous function f is shown. (Assume f ′ continues to ∞.) (b) At what value(s) of x does f have a local maximum? (Enter your answers as a comma-separated list.) At what value(s) of x does f have a local minimum? (Enter your answers as a comma-separated list
Use the graph of the derivative f ′ of a continuous function f is shown. (Assume f ′ continues to ∞.) (b) At what value(s) of x does f have a local maximum? (Enter your answers as a comma-separated list.) At what value(s) of x does f have a local minimum? (Enter your answers as a comma-separated list
Use the graph of the derivative f ′ of a continuous function f is shown. (Assume f ′ continues to ∞.)
(b) At what value(s) of x does f have a local maximum? (Enter your answers as a comma-separated list.)
At what value(s) of x does f have a local minimum? (Enter your answers as a comma-separated list.)
Transcribed Image Text:This is a graph of the function \( y = f'(x) \), which represents the derivative of a function \( f(x) \). The graph is plotted on a coordinate grid with the x-axis labeled from 0 to 10 and the y-axis labeled from -2 to 4.
Key features of the graph:
- The curve starts below the x-axis, indicating that the derivative is initially negative.
- Around \( x = 1 \), the curve crosses into positive territory, peaks near \( y = 2.5 \), and then dips slightly before peaking again at approximately \( x = 4.5 \) with a value just over 3.
- The graph then descends rapidly, crossing the x-axis near \( x = 5.5 \), reaching a local minimum below \( y = -1 \) close to \( x = 6.5 \).
- The curve rises again, crossing the x-axis near \( x = 8.5 \), and continues upward.
This graph showcases the behavior of the derivative of a function, indicating changes in the slope of \( f(x) \). Points where \( y = f'(x) = 0 \) correspond to local maxima and minima of \( f(x) \).
Formula Formula A function f ( x ) is also said to have attained a local minimum at x = a , if there exists a neighborhood ( a − δ , a + δ ) of a such that, f ( x ) > f ( a ) , ∀ x ∈ ( a − δ , a + δ ) , x ≠ a f ( x ) − f ( a ) > 0 , ∀ x ∈ ( a − δ , a + δ ) , x ≠ a In such a case f ( a ) is called the local minimum value of f ( x ) at x = a .
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