Question

the graph of ( y = f(x) ) is shown above. which of the following could be the graph of ( y = f(x) )?

Explanation:

Step1: Analyze the original function's behavior

The graph of \(y = f(x)\) has a maximum at \(x = 0\). By the first - derivative test, if a function \(y = f(x)\) has a local maximum at \(x=a\), then \(f^{\prime}(a)=0\) and \(f^{\prime}(x)\) changes sign from positive to negative as \(x\) passes through \(a\). Also, for \(|x|\) large enough, the function \(y = f(x)\) is decreasing (since the function approaches a horizontal asymptote from above as \(|x|\to\infty\)).

Step2: Analyze the sign of the derivative

For \(x\lt0\), the function \(y = f(x)\) is increasing. When a function \(y = f(x)\) is increasing on an interval \((-\infty,0)\), then \(f^{\prime}(x)>0\) for \(x\in(-\infty,0)\). For \(x > 0\), the function \(y = f(x)\) is decreasing. When a function \(y = f(x)\) is decreasing on an interval \((0,\infty)\), then \(f^{\prime}(x)<0\) for \(x\in(0,\infty)\)

Step3: Check the options

• Option A: The function in option A is increasing for \(x>0\) (since the slope is positive for \(x > 0\)), which is not consistent with \(y = f(x)\) being decreasing for \(x>0\).
• Option B: The function in option B has \(f^{\prime}(x)<0\) for \(x < 0\) (since the slope is negative for \(x<0\)), which is not consistent with \(y = f(x)\) being increasing for \(x < 0\).
• Option C: The function in option C has \(f^{\prime}(x)>0\) for \(x < 0\) (the function is above the \(x -\)axis for \(x < 0\)) and \(f^{\prime}(x)<0\) for \(x>0\) (the function is below the \(x -\)axis for \(x > 0\)) and \(f^{\prime}(0)=0\).
• Option D: The function in option D has \(f^{\prime}(x)>0\) for all \(x

eq0\) (the function is above the \(x -\)axis for \(x
eq0\)), which is not consistent with \(y = f(x)\) being decreasing for \(x>0\)

Answer:

C.