What is the maximum value of such that the graph of the parabola \(y = \dfrac{1}{3}x^2\) has at most one point of intersection with the line \(y = x+c?\)

Logic Jul 30, 2019

#1**+4 **

Assuming the question is "What is the maximum value of **c** such that. . ."

\(y\ =\ \frac13x^2\\~\\ y\ =\ x+c\)

Equate both expressions of y

\(\frac13x^2\ =\ x+c\\~\\ \frac13x^2-x-c\ =\ 0\)

This quadratic equation will have only one solution when the discriminant equals zero.

\((-1)^2-4(\frac13)(-c)\ =\ 0\\~\\ 1+\frac43c\ =\ 0\\~\\ \frac43c\ =\ -1\\~\\ c\ =\ -\frac34\)

hectictar Jul 30, 2019

#1**+4 **

Best Answer

Assuming the question is "What is the maximum value of **c** such that. . ."

\(y\ =\ \frac13x^2\\~\\ y\ =\ x+c\)

Equate both expressions of y

\(\frac13x^2\ =\ x+c\\~\\ \frac13x^2-x-c\ =\ 0\)

This quadratic equation will have only one solution when the discriminant equals zero.

\((-1)^2-4(\frac13)(-c)\ =\ 0\\~\\ 1+\frac43c\ =\ 0\\~\\ \frac43c\ =\ -1\\~\\ c\ =\ -\frac34\)

hectictar Jul 30, 2019