Q. 3 - Calculus

Question

Sketch the graph of a function f that is concave up everywhere. Then draw five tangent lines on the graph, and explain how you can see that the derivative of f is increasing.

Step-by-Step Solution

Answer

The function $f\left(x\right)=x^4$is zero for $x=0$ and f is concave up everywhere. The graph is as follows,

Step 1. Given information

Function f  is concave up everywhere.

Step 2. Derivative of the function.

Suppose the function is

$f\left(x\right)=x^4$

Now

$f^{\text{'}}\left(x\right)=4x^3$

And

$$\begin{gathered} f^{\text{'}\text{'}}\left(x\right)=4\left(3x^2\right) \\ =12x^2 \end{gathered}$$

Thus, $f\left(x\right)=0$ only when $x=0$, and f is positive to both the left and right of $x=0$. Similarly $f^{\text{'}\text{'}}\left(x\right)=0$ only when $x=0$, and $f^{\text{'}\text{'}}\left(x\right)$ is positive to both the left and right.

Step 3. Graphing the function

The graph is as follows,