Question

If \( f(x) = \int_0^x \left[ (a+1)(t+1)^2 - (a-1)(t^2 + t + 1) \right] dt \), then a possible positive value of \( a \), for which \( f'(x) = 0 \) has equal roots, is:

• A. 1
• B. -1
• C. 7
• D. 0

Hint:

For integrals with variable limits, use the fundamental theorem of calculus to differentiate. Ensure the discriminant of the quadratic equation is zero for equal roots.

Answer 1

The Correct Option is A

We are given the function: \[ f(x) = \int_0^x \left[ (a+1)(t+1)^2 - (a-1)(t^2 + t + 1) \right] dt \] To find \( f'(x) \), we differentiate the given integral with respect to \( x \). By the fundamental theorem of calculus: \[ f'(x) = (a+1)(x+1)^2 - (a-1)(x^2 + x + 1) \] For \( f'(x) = 0 \) to have equal roots, the discriminant of the quadratic equation must be zero. After simplifying and solving for \( a \), we find that the possible positive value of \( a \) is \( 1 \). Thus, the correct answer is option (1), \( a = 1 \).