Final answer:
To find the value of k that makes the function f(x) = 2x² - kx + 3 continuous, solve for k using the quadratic discriminant. The correct value of k is -3.
Step-by-step explanation:
To find the value of k that makes the function f(x) = 2x² - kx + 3 continuous, we need to solve for k. A function is continuous when there are no breaks or jumps in the graph. This means that the function has to be defined at every point, and the limit as x approaches any value must be equal to the value of the function at that point.
To find the value of k, we can use the fact that a quadratic function is continuous for all values of x if the discriminant (b² - 4ac) is greater than or equal to zero. In this case, the quadratic function is f(x) = 2x² - kx + 3, so we need to find the discriminant of this function and set it greater than or equal to zero.
The discriminant is given by b² - 4ac. Substituting the values from our function, we get (-k)² - 4(2)(3). Simplifying further, we get k² - 24. To make this greater than or equal to zero, k² - 24 ≥ 0. Solving for k, we find that k ≤ -√24 or k ≥ √24. Taking the square root of 24, we get k ≤ -4.9 or k ≥ 4.9. Therefore, the correct option is k = -3. Option d) k = -3 satisfies the condition for the function to be continuous.