Question

The graph of a function

Answer 1

(a) dy/dx is positive when the function is increasing
dy^2/dx^2 is positive when the graph is concave upward
Both are positive at point B
Answer: Point B

(b)  dy/dx is negative when the function is decreasing
dy^2/dx^2 is negative when the graph is concave downward
Both are negative at point E
Answer: Point E

(c) Answer: Point A

Answer 2

Note main properties of $\dfrac{dy}{dx}$ and $\dfrac{d^2y}{dx^2}:$  

1.  $\dfrac{dy}{dx}:$

• if  $\dfrac{dy}{dx}>0,$ the function is increasing;
• if  $\dfrac{dy}{dx}$ the function is decreasing;
• if  $\dfrac{dy}{dx}=0,$ the function can have minimum or maximum.

2.  $\dfrac{d^2y}{dx^2}:$

• if $\dfrac{d^2y}{dx^2}>0,$ the function is concave up;
• if $\dfrac{d^2y}{dx^2}$ the function is concave down;
• if $\dfrac{d^2y}{dx^2}=0,$ the function can have inflection point.

Now consider all options:

1. If $\dfrac{dy}{dx}>0,$ $\dfrac{d^2y}{dx^2}>0,$ then you should pick point B.

2. If $\dfrac{dy}{dx}$ $\dfrac{d^2y}{dx^2}$ then you should pick point E.

3. If $\dfrac{dy}{dx}$ $\dfrac{d^2y}{dx^2}>0,$ then you should pick point A.

Note that

• at point C $\dfrac{dy}{dx}>0,$ $\dfrac{d^2y}{dx^2}$
• at point D $\dfrac{dy}{dx}=0,$ $\dfrac{d^2y}{dx^2}$