k(x) = ax-4 when x is less than 2 and x^3 -2ax when x is greater than or equal to two . find a

To find the value of \( a \), we need to ensure the function \( k(x) \) is continuous at \( x = 2 \). This means the two expressions defining \( k(x) \) on different intervals must give the same result at \( x = 2 \). 

The function is given as:

\[
k(x) = \begin{cases} 
ax - 4 & \text{if } x < 2, \\
x^3 - 2ax & \text{if } x \geq 2.
\end{cases}
\]

For \( k(x) \) to be continuous at \( x = 2 \), the limit from the left must equal the limit from the right at \( x = 2 \).

### 1. Left-hand limit (\( x \to 2^- \)):

Using \( k(x) = ax - 4 \) for \( x < 2 \):
\[
\lim_{x \to 2^-} k(x) = a(2) - 4 = 2a - 4.
\]

### 2. Right-hand limit (\( x \to 2^+ \)):

Using \( k(x) = x^3 - 2ax \) for \( x \geq 2 \):
\[
\lim_{x \to 2^+} k(x) = 2^3 - 2a(2) = 8 - 4a.
\]

### 3. Set the limits equal to each other:

For continuity at \( x = 2 \), we set the two expressions equal to each other:
\[
2a - 4 = 8 - 4a.
\]

### 4. Solve for \( a \):

\[
2a + 4a = 8 + 4,
\]
\[
6a = 12,
\]
\[
a = 2.
\]

Thus, the value of \( a \) is \( \boxed{2} \).

Would you like more details, or do you have any questions?

Here are some related questions you might explore:
1. How would you verify if a function is differentiable at a point?
2. What happens if the left-hand and right-hand limits don't match at a point?
3. How do you solve continuity problems for piecewise functions in general?
4. What is the graphical interpretation of a discontinuity in a function?
5. How can you check if a function is continuous over its entire domain?

**Tip**: Continuity at a point means the function's value must equal the limit from both sides at that point!

k(x) = ax - 4 when x is less than 2 and x^3 - 2ax when x is greater than or equal to 2. Find a.

This page explains how to find the value of a parameter for a piecewise function to ensure continuity at a specific point. The problem involves calculating left-hand and right-hand limits, using concepts of limits and continuity. Suitable for Grades 11-12 students studying calculus or advanced algebra.

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