The line tangent to the graph of $f\left(x\right)\=x e^{-x}$ at any point $x\=b$ is

The difference between the tangent line and the function is to be no greater than $0.1$, so the following inequality  must be solved.

Type $f\left(x\right)\=\dots$, being sure to use the exponential "$e$".

Context Panel: Assign Function

Context Panel: Assign Function

Figure 3.4.3(a) can be obtained interactively by applying the Plot Builder to the expression

Figure 3.4.3(a) suggests two values of $b$.

Write the equation $\left|f\left(b\right)-Y\left(b\right)\right|\=0.1$
Press the Enter key.

Context Panel: Solve≻Numerically Solve

$\left|b{\ⅇ}^{-b}\+2{\ⅇ}^{-3}\left(b-3\right)-3{\ⅇ}^{-3}\right|\=0.1$

$5.221046886$

Write the equation $\left|f\left(b\right)-Y\left(b\right)\right|\=0.1$
Press the Enter key.

Context Panel:
Solve≻Numerically Solve from point≻$b\=0.1$

$\left|b{\ⅇ}^{-b}\+2{\ⅇ}^{-3}\left(b-3\right)-3{\ⅇ}^{-3}\right|\=0.1$

$0.4878338773$