if fx4xx use the power rule to find the derivative function

if f(x)=4/x+x use the power rule to find the derivative function f\'(x), determine the values of x at which the graph f(x) has a horozontal tangent line.

Solution

f(x) = 4/x + x

==> f(x) = 4x^-1 + x          since 1/aⁿ = a^-n

differentiating with respect to x

==> f \'(x) = 4(-1)x^-1-1 + 1x^1-1               since d/dx xⁿ = n x^n-1

==> f \'(x) = -4x^-2 + 1

==> f \'(x) = -(4/x²) + 1

Horizontal tangent line ==> f \'(x) = 0

==> -(4/x²) + 1 = 0

==> 4/x² = 1

==> x² = 4

==> x = 2 , -2

Hence the graph has horizontal tangent line at x = 2 , x = -2