Answer
D. tangent at x = 0 and and crosses at x = 3
Explanation
Given that:
[tex]f(x)=x^5-27x^2[/tex]
What to find:
To factor the given function and determine its behavior at the x-axis.
Step-by-step solution:
Solve for x, set f(x) equal to zero
[tex]\begin{gathered} x^5-27x^2=0 \\ \\ x^5=27x^2 \\ \\ Divide\text{ }both\text{ }sides\text{ }by\text{ }x^2 \\ \\ \frac{x^5}{x^2}=\frac{27x^2}{x^2} \\ \\ x^3=27 \\ \\ Take\text{ }cube\text{ }root\text{ }of\text{ }both\text{ }sides \\ \\ \sqrt[3]{x^3}=\sqrt[3]{27} \\ \\ x=3 \end{gathered}[/tex]
From f(x) = 0, therefore, x = 0
Also using a graphical method, we have:
Hence, the behavior of the x-axis is tangent at x = 0 and and crosses at x = 3.
The correct answer is option D. tangent at x = 0 and and crosses at x = 3
