# Calculus Problem

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• Mar 8th 2006, 02:54 PM
frozenflames
Calculus Problem
• Mar 9th 2006, 05:21 AM
CaptainBlack
a)
$
g(4)=\frac{f(1)+f(2)}{2}\times1+\frac{f(2)+f(4)}{2 }\times 2=2.5
$

$
g(-2)=\frac{f(1)+f(-2)}{2}\times (-3)=-6
$
• Mar 9th 2006, 05:26 AM
CaptainBlack
b)

Form the fundamental theorem of calculus:

$
\frac{d}{dx} \int_a^x f(t)dt=f(x)
$
,

so:

$
\frac{d}{dx} g|_{x=1}=f(1)=4
$
.

RonL
• Mar 9th 2006, 05:33 AM
CaptainBlack
c)

$g(x)$ is increasing from $-6$ at $x=-2$ untill $x=3$ from where it
decreases to $2.5$ at $x=4$.

Hence the mininmum of $g$ is $-6$ which occures when $x=-2$

RonL
• Mar 9th 2006, 06:21 AM
CaptainBlack
d)

x=1 is the only point of inflection the second derivative is positive to the
left of the point and negative to the right. At no other point is there a
change of sign of the second derivative and so there are no other points
of inflection.

RonL