the above graph is the graph of f′(x), then f(x) has a local maximum at x=___, a local minimum at x=____ and a critical point that is neither a local maximum nor a local minimum at x=____
the above graph is the graph of f′(x), then f(x) has a local maximum at x=___, a local minimum at x=____ and a critical point that is neither a local maximum nor a local minimum at x=____
the above graph is the graph of f′(x), then f(x) has a local maximum at x=___, a local minimum at x=____ and a critical point that is neither a local maximum nor a local minimum at x=____
If the above graph is the graph of f′(x), then f(x) has a local maximum at x=___, a local minimum at x=____
and a critical point that is neither a local maximum nor a local minimum at x=____
Transcribed Image Text:### Graph Explanation
This image is a graph of a polynomial function, showcasing its behavior on a Cartesian plane. The graph features:
- **Axes:**
- The horizontal axis (x-axis) ranges approximately from -10 to 10.
- The vertical axis (y-axis) ranges from -30 to 30.
- **Curve:**
- The blue curve represents the polynomial function.
- The function has two visible turning points:
- A local minimum occurs around x = -5, where the curve reaches its lowest point in that region.
- A local maximum approximately occurs around x = 0, indicating a peak before descending again.
- Another local minimum is observed around x = 4, where it again reaches a low point before ascending.
- **Behavior:**
- As x moves towards positive or negative infinity, the function appears to tend towards positive infinity, indicating the polynomial has a positive leading coefficient and an even degree.
- **Intercepts:**
- The curve crosses the y-axis in the negative region, around y = -6.
- The exact x-intercepts are not clear in the visible range, suggesting there may be no real roots or they lie outside the current view.
Such graphs are often used to study the characteristics and critical points of polynomial functions, assisting in understanding their behavior and applications in various contexts.
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 .
