Q. 4 - Calculus

Question

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

Step-by-Step Solution

Answer

The function $f\left(x\right)=-2x^4$is zero for $x=0$and $f\left(x\right)$is concave down for all left and right of $x=0.$

Step 1. Given information

Function f  is concave down everywhere 

Step 2. derivative of the function.

If function f and its derivative f' is differentiable on an interval I and second derivative f'' is negative then f'  is decreasing and f is concave down on interval I.

Consider a function $f\left(x\right)=-2x^4$

first derivative,

$f\text{'}\left(x\right)=-8x^3$

second derivative,

$f\text{'}\text{'}\left(x\right)=-24x^2$

$f\text{'}\text{'}\left(0\right)=0$ and $f\left(x\right)\text{'}\text{'}$ is negative for all left and right of $x=0.$

so $f\text{'}\left(0\right)=0$ and $f\text{'}\left(x\right)$is decreasing for all left and right of $x=0.$

 $f\left(0\right)=0$and $f\left(x\right)$is concave down for all left and right of $x=0.$

Step 3. The graph of the function.

Plot the graph of the function $f\left(x\right)=-2x^4$with five tangents.