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EEJ MAIN Mathematics QUESTION #969
Question 1
Let \(f(x) = \displaystyle\int_0^{x^2} \dfrac{t^2-8t+15}{e^t}\,dt,\ x \in \mathbb{R}\). Then the numbers of local maximum and local minimum points of \(f\), respectively, are:
  • 2 and 3✔️
  • 2 and 2
  • 3 and 2
  • 1 and 3
Correct Answer Explanation
\(f'(x)=\frac{x^4-8x^2+15}{e^{x^2}}\cdot 2x = \frac{2x(x^2-3)(x^2-5)}{e^{x^2}}\). Roots: \(x=0, \pm\sqrt{3}, \pm\sqrt{5}\). Analyzing sign changes: 2 local maxima (at \(\pm\sqrt{3}\)) wait — checking signs carefully gives 2 local maxima and 3 local minima.