Question

the graph of a function ( y = f(x) ) is shown: at what points are both ( \frac{dy}{dx} ) and ( \frac{d^{2}y}{dx^{2}} ) positive? select all that apply! b a e d c

Explanation:

Step1: Analyze the first - derivative ($\frac{dy}{dx}$)

The first - derivative $\frac{dy}{dx}$ represents the slope of the tangent line to the curve $y = f(x)$.

• If the function is increasing, $\frac{dy}{dx}>0$. Looking at the graph:
• At point $A$: The function is decreasing, so $\frac{dy}{dx}<0$.
• At point $B$: The function is increasing, so $\frac{dy}{dx}>0$.
• At point $C$: The function is increasing, so $\frac{dy}{dx}>0$.
• At point $D$: The slope of the tangent line is $0$ (it is a local maximum), so $\frac{dy}{dx} = 0$.
• At point $E$: The function is decreasing, so $\frac{dy}{dx}<0$.

Step2: Analyze the second - derivative ($\frac{d^{2}y}{dx^{2}}$)

The second - derivative $\frac{d^{2}y}{dx^{2}}$ represents the concavity of the function.

• If the function is concave up, $\frac{d^{2}y}{dx^{2}}>0$.
• At point $B$: The function is concave up (the graph is shaped like a "cup" opening upwards), so $\frac{d^{2}y}{dx^{2}}>0$.
• At point $C$: The function is concave down (the graph is shaped like a "cap" opening downwards), so $\frac{d^{2}y}{dx^{2}}<0$.

Answer:

B. Option Text