f(x) = {[kx^2, if x ≤ 2][3, if x > 2]continuous at x = 2.

Question

Determine the value of the constant k so that the function\(f(x) = \begin{cases} kx^2&, \quad if\, x ≤2\\3 &, \quad if\, x>2 \end{cases} \) continuous at x = 2.

Answer 1

Given : 

It is clear that when x < 2 and x > 2, the given function is continuous at x = 2. 

So, at x = 2

We know that, 

If f is continuous at x = c, then 

The Left–hand limit, the Right–hand limit and the value of the function at x = c exist and are equal to each other.

⇒ 4k = 3

⇒ k = \(\frac{3}{4}\)

Therefore, 

The required value of k is \(\frac{3}{4}\)