IIT-JEE 1999Question 1
The function $f(x)=\displaystyle\int_{-1}^{x}t\left(e^{t}-1\right)(t-1)(t-2)^{3}(t-3)^{5}\,dt$ has a local minimum at $x=$
Differentiation and its Application
Solution
Concept. $f'(x)$ is the integrand at $x$. Factors that never change sign can be ignored. Formulas used. (1) $f'$: $-\to+$ is a local minimumStep 1 - Derivative. $f'(x)=x(e^x-1)(x-1)(x-2)^3(x-3)^5$. Step 2 - Simplify the sign. $x(e^x-1)\ge0$ (both factors have the sign of $x$), so it does not change sign at $0$. $(x-2)^3$ and $(x-3)^5$ have the signs of $x-2$ and $x-3$. So $f'$ has the sign of $(x-1)(x-2)(x-3)$. Step 3 - Sign line.Step 4 - Classify. By (1), local minima at $x=1$ and $x=3$ (and a maximum at $x=2$). Answer: (B), (D)