Answer:
c = 6.25
Step-by-step explanation:
We are given the following piecewise function:
[tex]\left \{ {{cx^{2} + 2x, x < 3} \atop {3x^{3} - cx, x \geq 3}} \right[/tex]
Continuous function:
A function f(x) is continuous, at a point a, if:
[tex]\lim_{x \to a} f(x)[/tex] exists and [tex]\lim_{x \to a} f(x) = f(a)[/tex]
In this question:
Piece-wise function, so we have to verify if the limit exists.
The function changes at x = 3. So we verify at a = 3.
It will exist if:
[tex]\lim_{x \to 3^{-}} f(x) = \lim_{x \to 3^{+}} f(x)[/tex]
To the left:
Less than 3.
[tex]\lim_{x \to 3^{-}} f(x) = \lim_{x \to 3^{-}} cx^{2} + 2x = c*(3)^{2} + 2*3 = 9c + 6[/tex]
To the right:
Greater than 3.
[tex]\lim_{x \to 3^{+}} f(x) = \lim_{x \to 3^{+}} 3x^{3} - cx = 3*3^{3} - 3c = 81 - 3c[/tex]
f continuous:
They have to be equal:
[tex]\lim_{x \to 3^{-}} f(x) = \lim_{x \to 3^{+}} f(x)[/tex]
[tex]9c + 6 = 81 - 3c[/tex]
[tex]12c = 75[/tex]
[tex]c = \frac{75}{12}[/tex]
[tex]c = 6.25[/tex]
