Math Problem Statement
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
Solution
To find the value of , we need to ensure the function is continuous at . This means the two expressions defining on different intervals must give the same result at .
The function is given as:
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!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
Limits
Formulas
Left-hand limit: \lim_{x \to 2^-} k(x) = ax - 4
Right-hand limit: \lim_{x \to 2^+} k(x) = x^3 - 2ax
Continuity condition: Set left-hand limit = right-hand limit
Theorems
Continuity at a Point
Limits
Suitable Grade Level
Grades 11-12