Question

the graph of a twice - differentiable function f is shown in the figure above. which of the following is true?
a ( f(1)<f(1)<f(1) )
b ( f(1)<f(1)<f(1) )
c ( f(1)<f(1)<f(1) )
d ( f(1)<f(1)<f(1) )
e ( f(1)<f(1)<f(1) )

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.