Question

Consider the function \( h(x) = \frac{1}{4}x^4 - \frac{5}{3}x^3 + 3x^2 + 4x\).Find all values of x where the tangent line to the graph of y = h(x) is parallel to the line y = 4x + 3.

Answer 1

Consider the function \( h(x) = \frac{1}{4}x^4 - \frac{5}{3}x^3 + 3x^2 + 4x\).Find all values of x where the tangent line to the graph of y = h(x) is parallel to the line y = 4x + 3.

Answer 2

So y = 4x + 3 has a slope of 4

So h'(x) = 4

Answer 3

h'(x) = x^3 - 5x^2 + 6x + 4

and since h'(x) = 4

x^3 - 5x^2 + 6x + 4 = 4
x^3 - 5x^2 + 6x = 0

x(x^2 - 5x + 6) = 0

x = 0, 2, 3

Answer 4

\[\frac{ dh }{ dx }=4*\frac{ 1 }{ 4 }x^3-3*\frac{ 5 }{ 3 }x^2+6x+4=x^3-5x^2+6x+4\]
you are correct
find x

Answer 5

fortunately for you that is easy to solve, although it is usually hard to solve a cubic equatoin

Answer 6

Yeah, thanks all c: