The graph of the first derivative of a function is shown a) How can we know in which intervals the function f (x) is increasing and where it is decreasing? b) From f´ (x), how can we identify at what values of x, f (x) has a relative maximum or minimum?
The graph of the first derivative of a function is shown a) How can we know in which intervals the function f (x) is increasing and where it is decreasing? b) From f´ (x), how can we identify at what values of x, f (x) has a relative maximum or minimum?
The graph of the first derivative of a function is shown a) How can we know in which intervals the function f (x) is increasing and where it is decreasing? b) From f´ (x), how can we identify at what values of x, f (x) has a relative maximum or minimum?
The graph of the first derivative of a function is shown
a) How can we know in which intervals the function f (x) is increasing and where it is decreasing?
b) From f´ (x), how can we identify at what values of x, f (x) has a relative maximum or minimum?
Transcribed Image Text:The image is a graph of the function \( y = f'(x) \), which represents the derivative of a function.
### Description of the Graph:
- **Axes**: The graph has a horizontal \( x \)-axis and a vertical \( y \)-axis. Both axes are marked with numbers to indicate scale.
- **Curve**: The curve is a smooth, continuous wave that crosses the \( x \)-axis several times. It starts at a positive \( y \)-value, decreases through zero, reaches a negative peak, rises back through zero to a positive peak, and then declines again.
- **Key Points**:
- The graph intersects the \( x \)-axis at approximately \( x = 1 \), \( x = 4 \), and \( x = 7 \).
- It has a minimum point (negative peak) between \( x = 2 \) and \( x = 3 \).
- It reaches a maximum point (positive peak) between \( x = 5 \) and \( x = 6 \).
### Educational Explanation:
This graph illustrates how the derivative \( f'(x) \) of a function behaves over a certain interval on the \( x \)-axis. The points where the graph crosses the \( x \)-axis indicate where the original function \( f(x) \) has horizontal tangents, which are potential local maxima or minima. The peaks and troughs of \( f'(x) \) reveal changes in the slope of the original function, offering insights into its increasing or decreasing behavior. Understanding the derivative graph is crucial for comprehending the rate of change and the behavior of functions in calculus.
Formula Formula A function f(x) attains a local maximum 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 case, f(a) attains a local maximum value f(x) at x=a .
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