Question

at which of the five points on the graph in the figure at the right are $\frac{dy}{dx}$ and $\frac{d^{2}y}{dx^{2}}$ both negative?

Explanation:

Step1: Recall the meaning of the first derivative

The first derivative \(\frac{dy}{dx}\) represents the slope of the tangent line to the curve \(y = f(x)\). If \(\frac{dy}{dx}<0\), the function is decreasing.

Step2: Recall the meaning of the second derivative

The second derivative \(\frac{d^{2}y}{dx^{2}}\) represents the concavity of the curve. If \(\frac{d^{2}y}{dx^{2}}<0\), the curve is concave - down.

Step3: Analyze point A

At point A, the function is increasing (\(\frac{dy}{dx}>0\)), so it is not a candidate.

Step4: Analyze point B

At point B, the function is decreasing (\(\frac{dy}{dx}<0\)) and the curve is concave - down (\(\frac{d^{2}y}{dx^{2}}<0\)).

Step5: Analyze point C

At point C, the function is decreasing (\(\frac{dy}{dx}<0\)) but the curve is concave - up (\(\frac{d^{2}y}{dx^{2}}>0\)).

Step6: Analyze point D

At point D, the slope of the tangent line (first derivative) is approximately \(0\) (\(\frac{dy}{dx}\approx0\)).

Step7: Analyze point E

At point E, the function is increasing (\(\frac{dy}{dx}>0\)).

Answer:

B. Option Text