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Find all the points of local …

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Find all the points of local maxima and minima and the corresponding maximum and minimum values of the function f(x)=-3x^4/4 -8x^3 -45x^2/2 +105 ?
  • 1 answers

Sia ? 6 years ago

{tex}f'\left( x \right) = - 3{x^3} - 24{x^2} - 45x{/tex}

{tex} = - 3x\left( {{x^2} + 8x + 15} \right) = - 3x\left( {x + 5} \right)\left( {x + 3} \right){/tex}

{tex}f'\left( x \right) = 0 \Rightarrow x = - 5,x = - 3,x = 0{/tex}

{tex}f''\left( x \right) = - 9{x^2} - 48x - 45{/tex}

{tex}= - 3\left( {3{x^2} + 16x + 15} \right){/tex}

{tex}f''\left( 0 \right) = - 45 < 0{/tex}. Therefore, x = 0 is point of local maxima

{tex}f''\left( { - 3} \right) = 18 > 0{/tex}. Therefore, {tex}x = - 3{/tex} is point of local minima

{tex}f''\left( { - 5} \right) = - 30 < 0{/tex}. Therefore, {tex}x = - 5{/tex} is point of local maxima.

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