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Math Help - evaluatng others from extreme value of one function

  1. #1
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    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).
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  2. #2
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    Re: evaluatng others from extreme value of one function

    Hello, asifrahman1988!

    \text{Suppose }f(x)\text{ is differentiable everywhere.}
    \text{Assume that }f(x)\text{ has a local extreme value at the point }(2,0).
    \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.

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

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


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

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

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

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