Brief Calculus.

Question

Given f(x) find the values of x where the tangent line is horizontal.

f(x) = x²+5/3x-6

Answer 1

Please try to remember to use grouping symbols when using the "/" (forward slash symbol) to separate numerator and denominator of fractions.

Assuming this function is:

f(x) = (x²+5)/(3x-6), find f'(x) using the quotient rule, then determine when f'(x) equals 0. This works because f'(x) is a "formula" for slopes of tangent lines to f(x) and horizontal lines have a slope of 0.

Quotient Rule:

f'(x) = [(3x-6)(2x) - (x²+5)(3)]/(3x-6)²

= [6x²-12x - 3x²-15]/(3x-6)²

= (3x²-12x-15)/(3x-6)²

This equals zero when the numerator equals 0.

3x²-12x-15= 0

x²-4x-5=0 (divide all terms by 3)

(x-5)(x+1) = 0

Take it from there and check your work here:

desmos.com/calculator/bga0bm1sqw

In the above graph f'(x) is equivalent to g(x) -- confirm by enabling them to see the their graphs are identical.