**Problem Statement:** Sketch the graph of \( f(x) \). Choose the correct graph below.**Options:**- **Option A:** - Graph features a curve passing through the origin, starting from positive y-values and descending to negative y-values as x increases. It resembles a decreasing curve.- **Option B:** - Graph depicts a curve that starts from negative y-values, increases through the origin, and rises to positive y-values as x increases. It has an increasing trend and is selected as the correct graph.- **Option C:** - Graph displays a curve starting from positive y-values, decreasing through the origin, and continuing to fall as x increases. It follows a decreasing pattern.- **Option D:** - Curve begins at positive y-values, consistently decreases as x increases, and ends at lower negative y-values. It also shows a decreasing pattern.**Correct Answer:** Option B is selected as the correct graph. It represents a function that increases from negative y-values, passes through the origin, and continues to positive y-values. ### Analyzing the Graph of a Derivative#### Instructions:Use the graph of \( y = f'(x) \) to the right to discuss the graph of \( y = f(x) \). Organize your conclusions in a table, and sketch a possible graph of \( y = f(x) \).#### Graph Analysis:- The graph of \( y = f'(x) \) is depicted on a coordinate plane.- The x-axis ranges from -10 to 10 and the y-axis from -10 to 10.- The graph shows variations in \( f'(x) \) that affect the behavior of \( f(x) \).#### Table:| \( x \) | \( f'(x) \) | \( f(x) \) ||-----------------------|---------------------------|---------------------------------|| \( -\infty < x < -3 \) | Negative and increasing | Decreasing and concave upward || \( x = -3 \) | Local maximum | Inflection point || \( -3 < x < 0 \) | Positive and increasing | Decreasing and concave upward || \( x = 0 \) | Inflection point | Inflection point || \( 0 < x < 3 \) | Negative and decreasing | Increasing and concave upward || \( x = 3 \) | Local minimum | Inflection point || \( 3 < x < \infty \) | Positive and decreasing | Increasing and concave downward |#### Conclusion:- **Concavity and Monotonic Behavior**: The table summarizes how the sign and behavior of the derivative \( f'(x) \) influence the function \( f(x) \) over different intervals.- **Inflection Points**: Occur at \( x = -3, 0, 3 \) where the concavity of \( f(x) \) changes.- **Local Extrema**: The derivative's local maximum and minimum give insight into potential turning points on the graph of \( f(x) \).This analysis helps in sketching a possible graph of \( y = f(x) \) by understanding its intricate behavior over specified intervals.

f'(x) is always below x-axis, hence always negative.Since f'(x) is always negative then f(x) is always decreasing. When f'(x) magnitude decreases, f(x) changes slowly hence concave upward.When f'(x) magnitude increases, f(x) changes fastly hence concave downward.At local extrema where f''(x) = 0, f(x) has inflection points.When f'(x) = 0, f(x) has zero slope and hence horizontal. xf'(x) f(x)-∞<x<-3 negative and increasing decreasing and concave upwardx = -3 f'(x) = 0, local maxima or f''(x) = 0 zero slope, inflection point-3<x<0negative and decreasing decreasing and concave downwardx=0negative, f''(x) =0 decreasing, inflection point 0<x<3negative and increasing decreasing and concave upwardx=3 f'(x) = 0, local maxima or f''(x) = 0 zero slope, inflection point3<x<∞negative and decreasing decreasing and concave downwardoption D is correct.