Question

For what values of a and b is the line -2x + y = b tangent to the curve y=ax^3 when x=5? No decimal answers.

Answer 1

The slope of the line is 2
The slope of the cubic at x=5 is
\[ 3 a (5)^2 = 75 a\\
75 a = 2\\
a=\frac {75}2
\]

Answer 2

Now the line has to go through (5, 125 a) when x = 5 so you get
\[
b= -2(5) + 125 a=-10 + \frac { 125(75)}2+\frac{9355}{2}
\]

Answer 3

The last line should be
\[
b= -2(5) + 125 a=-10 + \frac { 125(75)}2=\frac{9355}{2}
\]

Answer 4

How did you get the 3a(5)^2?=75a?

Answer 5

The drivative of a x^3 is 3 a x^2 at 5 is 3 (a)(5)^2

Answer 6

I don't understand what's going on. I understood the thing earlier, but I don't know where these derivatives came from and how you got them? All the question was asking for was for the values of a and b, and we got them. What do I do with the derivatives?