First slide

Evaluation of definite integrals

Question

The inflection points on the graph of function $y = \int_{0}^{x} (t - 1) (t - 2)^{2} d t$ are

Moderate

A

$x = - 1$
B

$x = 3 / 2$
C

$x = 4 / 3$
D

$x = 1$

Solution

$\frac{d y}{d x} = (x - 1) (x - 2)^{2}$ so $\frac{d^{2} y}{d x^{2}} = (x - 2) (3 x - 4) .$
The points of inflection are given by $\frac{d^{2} y}{d x^{2}} = 0$ so $x = 2, x = 4 / 3$
are points of inflection.

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