Questions: Consider the function f(x) graphed below. For this function, are the following nonzero quantities positive or negative? (For each, enter the word positive or negative) f(3) is f'(3) is f''(3) is

Consider the function f(x) graphed below. For this function, are the following nonzero quantities positive or negative? (For each, enter the word positive or negative) f(3) is f'(3) is f''(3) is

Transcript text: Consider the function $f(x)$ graphed below. For this function, are the following nonzero quantities positive or negative? (For each, enter the word positive or negative) $f(3)$ is $\square$ $f^{\prime}(3)$ is $\square$ $f^{\prime \prime}(3)$ is $\square$

Solution

Solution Steps

Step 1: Determine the sign of f(3)

The graph plots _y_ = f(_x_). At _x_ = 3, the graph is above the _x_-axis, so _y_ = f(3) is positive.

Step 2: Determine the sign of f'(3)

The first derivative, f'(_x_), corresponds to the slope of the tangent line. At _x_ = 3, the function is decreasing, meaning the slope of the tangent line is negative. Thus f'(3) is negative.

Step 3: Determine the sign of f''(3)

The second derivative, f''(_x_), corresponds to the concavity of the function. At _x_ = 3, the graph is concave down, which means the second derivative is negative. Thus, f''(3) is negative.

Final Answer:

f(3) is positive f'(3) is negative f''(3) is negative

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