cmi-entrance 2019 QB6

cmi-entrance · India · ugmath 10 marks Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions

(a) Compute $\dfrac{d}{dx}\left[\int_{0}^{e^{x}} \log(t)\cos^{4}(t)\,dt\right]$.
(b) For $x > 0$ define $F(x) = \int_{1}^{x} t\log(t)\,dt$.
i. Determine the open interval(s) (if any) where $F(x)$ is decreasing and the open interval(s) (if any) where $F(x)$ is increasing.
ii. Determine all the local minima of $F(x)$ (if any) and the local maxima of $F(x)$ (if any).

(a) Compute $\dfrac{d}{dx}\left[\int_{0}^{e^{x}} \log(t)\cos^{4}(t)\,dt\right]$.

(b) For $x > 0$ define $F(x) = \int_{1}^{x} t\log(t)\,dt$.

i. Determine the open interval(s) (if any) where $F(x)$ is decreasing and the open interval(s) (if any) where $F(x)$ is increasing.

ii. Determine all the local minima of $F(x)$ (if any) and the local maxima of $F(x)$ (if any).

Paper Questions

QA1 QA10 QA2 QA3 QA4 QA5 QA6 QA7 QA8 QA9 QB1 QB2 QB3 QB4 QB5 QB6