Question

consider the graph of ( g(x) ) shown below.
(a) if ( g(x) ) is the first derivative of ( f(x) ), what is the nature of ( f(x) ) when ( x = 1 )?
(b) if ( g(x) ) is the second derivative of ( f(x) ), what is the nature of ( f(x) ) when ( x = 4 )?
(a) if ( g(x) ) is the first derivative of ( f(x) ), what is the nature of ( f(x) ) when ( x = 1 )?
a. ( f(x) ) has a local maximum at ( x = 1 ).
b. ( f(x) ) is decreasing at ( x = 1 ).
c. ( f(x) ) is increasing at ( x = 1 ).
d. ( f(x) ) has a local minimum at ( x = 1 ).
(b) if ( g(x) ) is the second derivative of ( f(x) ), what is the nature of ( f(x) ) when ( x = 4 )?
a. ( f(x) ) is concave down at ( x = 4 ).
b. ( f(x) ) has an inflection point at ( x = 4 ).
c. ( f(x) ) is concave up at ( x = 4 ).

Explanation:

(a)

Step1: Recall the relationship between \(f(x)\) and its first - derivative \(g(x)=f^{\prime}(x)\)

If \(g(x)=f^{\prime}(x)\), when \(f^{\prime}(x)>0\), \(f(x)\) is increasing; when \(f^{\prime}(x)<0\), \(f(x)\) is decreasing.
From the graph, when \(x = 1\), \(g(1)=f^{\prime}(1)>0\) (the \(y\) - value of \(g(x)\) at \(x = 1\) is positive).

(b)

Step1: Recall the relationship between \(f(x)\) and its second - derivative \(g(x)=f^{\prime\prime}(x)\)

If \(g(x)=f^{\prime\prime}(x)\), when \(f^{\prime\prime}(x)>0\), \(f(x)\) is concave up; when \(f^{\prime\prime}(x)<0\), \(f(x)\) is concave down.
From the graph, when \(x = 4\), \(g(4)=f^{\prime\prime}(4)<0\) (the \(y\) - value of \(g(x)\) at \(x = 4\) is negative).

Answer:

(a) C. \(f(x)\) is increasing at \(x = 1\).
(b) A. \(f(x)\) is concave down at \(x = 4\).