Step 1: Understand the Condition for 3 Distinct Real Roots
Let the given function be:
$$f(x) = 2x^3 - 3x^2 - 12x + k$$
For a cubic polynomial with a positive leading coefficient ($$2 > 0$$) to intersect the x-axis at three distinct points, its graph must rise to a local maximum above the x-axis ($$f(x_{\text{max}}) > 0$$) and dip to a local minimum below the x-axis ($$f(x_{\text{min}}) < 0$$).
Mathematically, this translates to the condition:
$$f(x_{\text{max}}) \cdot f(x_{\text{min}}) < 0$$
Step 2: Find the Critical Points
To find the turning points, differentiate $$f(x)$$ with respect to $$x$$ and equate it to zero:
$$f'(x) = 6x^2 - 6x - 12$$
Set $$f'(x) = 0$$:
$$6(x^2 - x - 2) = 0$$ $$(x - 2)(x + 1) = 0$$
The critical points are $$x = -1$$ and $$x = 2$$.
- By the first derivative test (or looking at the sign change of $$f'(x)$$), $$x = -1$$ is the point of local maxima.
- $$x = 2$$ is the point of local minima.
Step 3: Evaluate the Function at Critical Points
Substitute these critical points back into original function $f(x)$:
- At $$x = -1$$ (Local Maxima):$$f(-1) = 2(-1)^3 - 3(-1)^2 - 12(-1) + k$$ $$f(-1) = -2 - 3 + 12 + k = k + 7$$
- At $$x = 2$$ (Local Minima):$$f(2) = 2(2)^3 - 3(2)^2 - 12(2) + k$$ $$f(2) = 16 - 12 - 24 + k = k - 20$$
Step 4: Apply the Root Condition
For three distinct real roots:
$$(k + 7)(k - 20) < 0$$
Using the method of intervals (wavy curve method), the range of $$k$$ is:
$$-7 < k < 20$$
Step 5: Count the Integral Values
We need to find the number of integers strictly between $$-7$$ and $$20$$:
$$\text{Number of integers} = 20 - (-7) - 1 = 26$$
The specific integers are from $$-6$$ to $$19$$ inclusive.
Correct Option: B. 26
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