Show that there is no value of x for which f(x) = -1/x and g(x)= -x^5 have tangents with the same slope.

Question

Chapter 2 of Calculus Hey i’ve got this homework question in which i’m having trouble with

Answer 1

Hey Jake, a derivative tells you the slope of the tangent line at any point. Take the derivative of each function and see if f'(x) ever equals g'(x)

Answer 2

The derivative gives you the value of slope of a tangent line in any function.

So we have f(x) = -1/x and g(x) = -x^5

Now let's take derivative of each function to find the slope of the tangent line.

f ' (x) = 1/(x^2) and g ' (x) = -5x^4

Now let's plug and try some values of x in both derivatives i.e. the slope of the tangent lines [ f ' (x) and g ' (x) ]

When x = 1, f ' (x) = 1 and g ' (x) = -5

when x = 2, f ' (x) = 1/4 and g ' (x) = -80

So we can see that there is no value of x such that the slopes of two tangent lines are same.

Answer 3

The slope of f(x) = f ' (x) = 1/x²

The slope of g(x) = g ' (x) = -5x⁴

If the slopes are the same then:

1/x² = -5x⁴

1 = -5x⁶

-1/5 = x⁶

There are no real number solutions to this (even number roots result in imaginary number answers). so there is no real number value of x for which the slopes are the same.