the graph of a twice - differentiable function f is shown in the figure above. which of the following is…

the graph of a twice - differentiable function f is shown in the figure above. which of the following is true?\na ( f(1)<f(1)<f(1) )\nb ( f(1)<f(1)<f(1) )\nc ( f(1)<f(1)<f(1) )\nd ( f(1)<f(1)<f(1) )\ne ( f(1)<f(1)<f(1) )

the graph of a twice - differentiable function f is shown in the figure above. which of the following is true?\na ( f(1)<f(1)<f(1) )\nb ( f(1)<f(1)<f(1) )\nc ( f(1)<f(1)<f(1) )\nd ( f(1)<f(1)<f(1) )\ne ( f(1)<f(1)<f(1) )

Answer

Explanation:

Step1: Analyze ( f(1) )

From the graph, when ( x = 1 ), ( y=f(1)=0 ).

Step2: Analyze ( f^{\prime}(1) )

The first - derivative ( f^{\prime}(1) ) represents the slope of the tangent line to the graph of ( y = f(x) ) at ( x = 1 ). The tangent line at ( x = 1 ) has a positive slope, so ( f^{\prime}(1)>0 ).

Step3: Analyze ( f^{\prime\prime}(1) )

The second - derivative ( f^{\prime\prime}(x) ) represents the concavity of the function. The graph of ( y = f(x) ) is concave down at ( x = 1 ) (since the slope of the tangent line is decreasing as ( x ) increases), so ( f^{\prime\prime}(1)<0 ).

Answer:

( f^{\prime\prime}(1)<f(1)<f^{\prime}(1) ), so the answer is D.