Let the function, f(x)={−3ax2−2,a2+bx,x<1x⩾1 be differentiable for all x∈R, where a>1, b∈R. If the area of the region enclosed by y=f(x) and the line y=−20 is α+β3, α,β∈Z, then the value of α+β is:
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Step-by-Step Solution
Step 1: Apply Differentiability Conditions
For a function to be differentiable at a point, it must first be continuous at that point. This means the left-hand limit and the right-hand limit of the function must be equal to the function's value at that point. Additionally, the left-hand derivative must equal the right-hand derivative at that point.
Step 2: Solve for 'a' and 'b'
By applying the continuity condition, we equate the two parts of the function at x=1. Then, we find the derivatives of each part and equate them at x=1. Substituting the value of b from the derivative equation into the continuity equation, we get a quadratic equation in terms of a. We solve this quadratic equation for a.
Step 3: Determine 'a' and 'b' values
Using the quadratic formula, we find two possible values for a. Since the problem states that a>1, we select the positive root. Then, we substitute this value of a back into the equation for b to find its value.
Step 4: Find intersection points with y=−20
To find the area, we need the intersection points of y=f(x) with y=−20. We set each part of f(x) equal to −20 and solve for x. For the first part, x<1, we find the x -values where −3ax2−2=−20. For the second part, x≥1, we find the x -value where a2+bx=−20. We rationalize the denominator for the x values from the first part.
Step 5: Calculate the Area
We identify the relevant intersection points. The area is calculated by integrating the difference between f(x) and y=−20 over the appropriate intervals. The first integral is from x1 to 1 for the parabolic part, and the second integral is from 1 to x2 for the linear part. After performing the integration and substituting the limits, we find the total area.
Step 6: Determine α and β
Comparing the calculated area with the given form α+β3, we can identify the values of α and β.
Step 7: Calculate α+β
Finally, we sum the values of α and β to get the required result.