Question

1 mark for review
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, the function \( f(x) \) passes through the point \( (1, 0) \) (since it intersects the \( x \)-axis at \( x = 1 \)), so \( f(1)=0 \).

Step2: Analyze \( f'(1) \)

The first derivative \( f'(x) \) represents the slope of the tangent line to the graph of \( f(x) \) at a point \( x \). At \( x = 1 \), the graph of \( f(x) \) is increasing (since it goes from below the \( x \)-axis to above as \( x \) increases through 1), so the slope of the tangent line at \( x = 1 \) is positive. Thus, \( f'(1)>0 \).

Step3: Analyze \( f''(1) \)

The second derivative \( f''(x) \) represents the concavity of the function \( f(x) \). The graph of \( f(x) \) is concave down (since it is increasing at a decreasing rate, the curve is curving downward), so \( f''(x)<0 \) for all \( x \) in the domain shown, including \( x = 1 \). Thus, \( f''(1)<0 \).

Step4: Compare the values

We have \( f''(1)<0 \), \( f(1) = 0 \), and \( f'(1)>0 \). So the order is \( f''(1)<f(1)<f'(1) \).

Answer:

D. \( f''(1)<f(1)<f'(1) \)