First slide

Evaluation of definite integrals

Question

For $x \in (0, 5 π / 2)$ , define $f (x) = \int_{0}^{x} \sqrt{t} \sin t d t$ Then $f$ has

Moderate

A

local maximum at $π$ and local minimum at $2 π$ .
B

local maximum at $π$ and $2 π$
C

local minimum at $π$ and $2 π$
D

local minimum at $π$ and local maximum at $2 π$

Solution

$f^{'} (x) = \sqrt{x} \sin x$ , so $f^{'} (x) = 0$
$\Rightarrow x = π, 2 π$ but $f^{''} (x) = \sqrt{x} \cos x - \frac{1}{2 \sqrt{x}} \sin x,$ so $f^{''} (π) =$
$- \sqrt{π} < 0$ and $f^{''} (2 π) = \sqrt{2 π} > 0$ . Hence $f$ has local maximum
at $x = π$ and local minimum at $x = 2 π$ .

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