Question

at the point shown on the function above, which of the following is true?
\\( \bigcirc f^{\prime}<0, f^{\prime \prime}>0 \\)
\\( \bigcirc f^{\prime}<0, f^{\prime \prime}<0 \\)
\\( \bigcirc f^{\prime}>0, f^{\prime \prime}<0 \\)
\\( \bigcirc f^{\prime}>0, f^{\prime \prime}>0 \\)
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Explanation:

Step1: Analyze the first - derivative \(f^{\prime}\)

The first - derivative \(f^{\prime}(x)\) represents the slope of the tangent line to the function \(y = f(x)\). At the given point, the function is increasing. When a function \(y = f(x)\) is increasing, the slope of the tangent line is positive. So, \(f^{\prime}>0\).

Step2: Analyze the second - derivative \(f^{\prime\prime}\)

The second - derivative \(f^{\prime\prime}(x)\) represents the concavity of the function. If the function \(y = f(x)\) is concave down, then \(f^{\prime\prime}(x)<0\). At the given point, the function is concave down (the graph “holds water” in an upside - down sense).

Answer:

\(f^{\prime}>0,f^{\prime\prime}<0\) (the third option)