Answer to Question #100202 in Differential Equations for Beast

Question

For what value of x is the tangent of the graph of the function f (x) = x4− ax 3−4 parallel to the x axis?

Answer 1

From the geometric interpretation of derivative it follows that a tangent line to the graph of the function $f(x)$ at the point $(x_0,f(x_0))$ is parallel to the $x$ axis if, and only if $f'(x_0) = 0.$

Then, we have the equation

$f'(x) = (x^4 - ax^3 - 4)' = 4x^3 - 3ax^2 = x^2(4x - 3a) = 0$

It has two solutions: $x_1 = 0,\; x_2 = \frac{3}{4}a.$

Answer: Either $x=0$ or $x = \frac{3}{4}a.$