# Thread: evaluatng others from extreme value of one function

1. ## evaluatng others from extreme value of one function

Suppose f is differentiable on -infinity and +infinity(everywhere) and assume that it has a local extreme value at the point (2,0). Let g(x)=xf(x)+1 and h(x)=xf(x)+x+1 for all values of x.
Evaluate g(2), h(2), g'(2) and h'(2).

2. ## Re: evaluatng others from extreme value of one function

Hello, asifrahman1988!

$\displaystyle \text{Suppose }f(x)\text{ is differentiable everywhere.}$
$\displaystyle \text{Assume that }f(x)\text{ has a local extreme value at the point }(2,0).$
$\displaystyle \text{Let }g(x)\:=\:x\!\cdot\!f(x)+1\,\text{ and }\,h(x)\:=\:x\!\cdot\!f(x)+x+1\,\text{ for all values of }x.$

$\displaystyle \text{Evaluate: }g(2),\;h(2),\;g'(2),\;h'(2)$

We are given some information . . .
. . $\displaystyle (2,0)\text{ is on the graph: }\:f(2) = 0$
. . $\displaystyle \text{The first derivative is 0 when }x=2\!:\;f'(2) \,=\,0$

$\displaystyle g(x) \;=\;x\!\cdot\!f(x) + 1$
$\displaystyle g(2) \;=\;2\!\cdot\!f(2) + 1 \;=\;2\!\cdot\!0 + 1 \;=\;1$

$\displaystyle h(x) \;=\;x\!\cdot\!f(x) + x + 1$
$\displaystyle h(2) \;=\;2\!\cdot\!f(2) + 2 + 1 \;=\;2\!\cdot\!0 + 2 + 1 \;=\;3$

$\displaystyle g'(x) \;=\;x\!\cdot\!f'(x) + f(x)$
$\displaystyle g'(2) \;=\;2\!\cdot\!f'(2) + f(2) \;=\;2\!\cdot\!0 + 0 \;=\;0$

$\displaystyle h'(x) \;=\;x\!\cdot\!f'(x) + f(x) + 1$
$\displaystyle h'(2) \;=\;2\!\cdot\!f'(2) + f(2) + 1 \;=\;2\!\cdot\!0+ 0 + 1 ;=\;1$