Question

Sketch the graph of a function whose first derivative is always negative and second derivative is always positive.

Answer 1

The function is always decreasing (first derivative always negative) and the curve is concave upward (second derivative always positive).

A function whose first derivative is always negative and second derivative is always positive will be decreasing at all points and concave upward. One example of such a function is a concave up, decreasing quadratic function.

Consider the function:

f(x)=−x^2

Let's analyze its derivatives:

First Derivative:

(x)=−2x

The first derivative is always negative for any real value of

x, indicating that the function is always decreasing.

Second Derivative:

(x)= 2

The second derivative is a constant (2), which is always negative. This implies that the function is concave upward at all points.

Now, let's sketch the graph of f(x)=−x^2 :

In this graph, you can see that the function is always decreasing (first derivative always negative) and the curve is concave upward (second derivative always positive).

Answer 2

y = f(x)
y' < 0
y'' > 0

Function must be decreasing and concave


`f(x)= e^(-x) \\ \\f'(x)=- e^(-x)\ \textless \ 0=-(1)/(e^x) \ \textless \ 0 \\ \\f''(x)=(-e^(-x))'=e^(-x)= (1)/(e^x) \ \textgreater \ 0`