# Calculus Problem

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

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

Form the fundamental theorem of calculus:

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

so:

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

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

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

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

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