How to answer these questions using differentiation ?

A curve is defined by #y = x ^ 3 - 12x#

a) the value of x where y is maximal,
b) the touch line equation to the curve at point (3, -9)

1 Answer
Jim G.
Mar 30, 2018

#x=-2" and "y=15x-54#

Explanation:

#(a)#

#"to find maxima differentiate and equate to zero"#

#rArrdy/dx=3x^2-12=0#

#rArr3(x-2)(x+2)=0rArrx=+-2#

#"to determine max/min use the "color(red)"second derivative test"#

#• " If "(d^2y)/(dx^2)>0" at x = a then minimum"#

#• " If "(d^2y)/(dx^2)<0" at x = a then maximum"#

#(d^2y)/(dx^2)=6x#

#x=2to(d^2y)/(dx^2)=12>0rArr"minimum"#

#x=-2to(d^2y)/(dx^2)=-12<0rArr" maximum"#

#rArr"y is a maximum when "x=-2#

#(b)#

#•color(white)(x)m_(color(red)"tangent")=dy/dx" at x = 3"#

#x=3tody/dx=27-12=15#

#rArry+9=15(x-3)larrcolor(blue)"point-slope form"#

#rArry=15x-45-9#

#rArry=15x-54larrcolor(red)"equation of tangent"#

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