Question

1. using the figure below, complete the chart by indicating whether each value is positive (+), negative (-), or zero (0) at the indicated points. for these problems, if the point appears to be a max or min, assume it is. if it appears to be a point of inflection, assume it is.

Explanation:

Step1: Analyze \(f(x)\)

- For \(f(x)\):
- If the point is above the \(x -\)axis, \(f(x)>0\) (positive, \(+\)).
- If the point is below the \(x -\)axis, \(f(x)<0\) (negative, \(-\)).
- If the point is on the \(x -\)axis, \(f(x) = 0\).
- Point \(a\) is below the \(x -\)axis, so \(f(a)<0\).
- Point \(b\) is above the \(x -\)axis, so \(f(b)>0\).
- Point \(c\) is above the \(x -\)axis, so \(f(c)>0\).
- Point \(d\) is on the \(x -\)axis, so \(f(d)=0\).
- Point \(e\) is above the \(x -\)axis, so \(f(e)>0\).
- Point \(f\) is above the \(x -\)axis, so \(f(f)>0\).
- Point \(g\) is above the \(x -\)axis, so \(f(g)>0\).
- Point \(h\) is below the \(x -\)axis, so \(f(h)<0\).
- Point \(i\) is below the \(x -\)axis, so \(f(i)<0\).
- Point \(j\) is on the \(x -\)axis, so \(f(j)=0\).

Step2: Analyze \(f^{\prime}(x)\)

- The first - derivative \(f^{\prime}(x)\) represents the slope of the tangent line to the curve \(y = f(x)\).
- If the function is increasing (\(y\) - values increase as \(x\) - values increase), \(f^{\prime}(x)>0\) (positive, \(+\)).
- If the function is decreasing (\(y\) - values decrease as \(x\) - values increase), \(f^{\prime}(x)<0\) (negative, \(-\)).
- If the function has a local maximum or minimum, \(f^{\prime}(x) = 0\).
- At point \(a\): The function is increasing, so \(f^{\prime}(a)>0\).
- At point \(b\): The function has a local maximum, so \(f^{\prime}(b)=0\).
- At point \(c\): The function is decreasing, so \(f^{\prime}(c)<0\).
- At point \(d\): The function has a local minimum, so \(f^{\prime}(d)=0\).
- At point \(e\): The function is increasing, so \(f^{\prime}(e)>0\).
- At point \(f\): The function has a local maximum, so \(f^{\prime}(f)=0\).
- At point \(g\): The function is decreasing, so \(f^{\prime}(g)<0\).
- At point \(h\): The function has a local minimum, so \(f^{\prime}(h)=0\).
- At point \(i\): The function is increasing, so \(f^{\prime}(i)>0\).
- At point \(j\): The function is increasing (since the slope of the tangent line at \(j\) (a non - critical point on the increasing part of the linear - like behavior near \(j\)) is positive, \(f^{\prime}(j)>0\).

Step3: Analyze \(f^{\prime\prime}(x)\)

- The second - derivative \(f^{\prime\prime}(x)\) represents the concavity of the function.
- If the function is concave up (\(y^{\prime}\) is increasing), \(f^{\prime\prime}(x)>0\) (positive, \(+\)).
- If the function is concave down (\(y^{\prime}\) is decreasing), \(f^{\prime\prime}(x)<0\) (negative, \(-\)).
- If the function has an inflection point, \(f^{\prime\prime}(x) = 0\).
- At point \(a\): The function is concave down (the slope of the tangent is decreasing as \(x\) increases), so \(f^{\prime\prime}(a)<0\).
- At point \(b\): The function is concave down (since the function has a local maximum and is concave down around it), so \(f^{\prime\prime}(b)<0\).
- At point \(c\): The function has an inflection point (the concavity changes), so \(f^{\prime\prime}(c)=0\).
- At point \(d\): The function is concave up (the slope of the tangent is increasing as \(x\) increases), so \(f^{\prime\prime}(d)>0\).
- At point \(e\): The function has an inflection point (the concavity changes), so \(f^{\prime\prime}(e)=0\).
- At point \(f\): The function is concave down (the slope of the tangent is decreasing as \(x\) increases), so \(f^{\prime\prime}(f)<0\).
- At point \(g\): The function is concave down (the slope of the tangent is decreasing as \(x\) increases), so \(f^{\prime\prime}(g)<0\).
- At point \(h\): The function is concave up (the slope of the tangent is increasing as \(x\) increases), so \(f^{\prime\prime}(h)>0\).
- At point \(i\): The function is concave up (the slope of the tangent is increasing as \(x\) increases), so \(f^{\prime\prime}(i)>0\).
- At point \(j\): The function is concave up (assuming the linear - like behavior near \(j\) is part of a concave - up curve), so \(f^{\prime\prime}(j)>0\).

Answer:

\(x\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)	\(f\)	\(g\)	\(h\)	\(i\)	\(j\)
\(f^{\prime}(x)\)	\(+\)	\(0\)	\(-\)	\(0\)	\(+\)	\(0\)	\(-\)	\(0\)	\(+\)	\(+\)
\(f^{\prime\prime}(x)\)	\(-\)	\(-\)	\(0\)	\(+\)	\(0\)	\(-\)	\(-\)	\(+\)	\(+\)	\(+\)