Question

assume ( f(x) ) is continuous on ( (-infty,infty) ). use the given information to sketch the graph of ( f ).

choose the correct graph below.

Explanation:

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

• When \(f^{\prime}(x)>0\), the function \(f(x)\) is increasing.
• When \(f^{\prime}(x) = 0\), the function \(f(x)\) has critical points. At \(x = 2\) and \(x=5\), \(f^{\prime}(x)=0\).
• When \(f^{\prime}(x)\) has a non - differentiable (ND) point (at \(x = 0\) for \(f^{\prime}(x)\)), the slope of the function changes in a non - smooth way.
• For \(x<0\), \(f^{\prime}(x)>0\) (function is increasing), for \(05\), \(f^{\prime}(x)>0\) (function is increasing).

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

• When \(f^{\prime\prime}(x)>0\), the function \(f(x)\) is concave up.
• When \(f^{\prime\prime}(x)<0\), the function \(f(x)\) is concave down.
• At \(x = 3\), \(f^{\prime\prime}(x)=0\) (inflection point). For \(x<0\), \(f^{\prime\prime}(x)>0\) (concave up), for \(03\), \(f^{\prime\prime}(x)>0\) (concave up).

Step3: Use the function values

• \(f(-4)=-3\), \(f(0) = 0\), \(f(2)=5\), \(f(3)=3\), \(f(5)=-2\), \(f(6)=0\)

Answer:

Option A.