Question

find all values of x where the tangent lines to y = x^8 and y = x^9 are parallel. if there is more than one x - value, enter them in a comma - separated list. if an x - value does not exist, enter dne.

Explanation:

Step1: Find the derivatives

The derivative of $y = x^{8}$ using the power - rule $(x^n)'=nx^{n - 1}$ is $y'=8x^{7}$. The derivative of $y = x^{9}$ is $y'=9x^{8}$.

Step2: Set the derivatives equal

Since parallel tangent lines have equal slopes, we set $8x^{7}=9x^{8}$.

Step3: Rearrange the equation

Move all terms to one side: $9x^{8}-8x^{7}=0$. Factor out $x^{7}$: $x^{7}(9x - 8)=0$.

Step4: Solve for x

Using the zero - product property, if $ab = 0$, then either $a = 0$ or $b = 0$.
For $x^{7}=0$, we get $x = 0$. For $9x-8=0$, we solve for $x$: $9x=8$, so $x=\frac{8}{9}$.

Answer:

$0,\frac{8}{9}$