+1206 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 \(y = x+c?\)
+9465 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.
\((-1)^2-4(\frac13)(-c)\ =\ 0\\~\\ 1+\frac43c\ =\ 0\\~\\ \frac43c\ =\ -1\\~\\ c\ =\ -\frac34\)
+9465 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.
\((-1)^2-4(\frac13)(-c)\ =\ 0\\~\\ 1+\frac43c\ =\ 0\\~\\ \frac43c\ =\ -1\\~\\ c\ =\ -\frac34\)
