Find the two points where the tangent line is parallel to the line

Question

The tangent line to the curve y=(1/3)x^3-4x^2+17x+24 is parallel to the line 25x-5y=2 at two points on the curve. Find the two points.

Answer 1

QUESTION

The tangent line to the curve y=(1/3)x^3-4x^2+17x+24 is parallel to the line 25x-5y=2 at two points on the curve. Find the two points.

y'=x²-8x+17 is the tangent for y which needs to have the slope of 25x-5y=2. 5y=25x-2 so slope is 25/5=5.

x²-8x+17=5 at the two points, x²-8x+12=(x-6)(x-2). The x values for the points are 2 and 6.

The y values are correspondingly 8/3-16+34+24=134/3 and 72-144+102+24=54.

The points are (2,134/3) and (6,54).