IIT JEE , AIEEE Forum - Mathematics , Differential Calculus - RD SHARMA..find the condition such that x3+ax2+bx+c has a local maxima at certain x? R+

killerkiran

RD SHARMA..find the condition such that x3+ax2+bx+c has a local maxima at certain x? R+

pink_ele

for local maxima ....
3x^2+2ax+b=0
shud have real roots ...........i.e...4a^2-12b>0
a^2>3b
such that ........6x+2a3x+2a<0
where x is d root ......

killerkiran

but what is the condition such that it forms at certain x belong to R+

spideyunlimited

Find the condition such that x^3+ax^2+bx+c has a local maxima at certain x E R+

f(x) = x^3+ax^2+bx+c
f'(x) = 3x^2 + 2ax + b = 0

x = -2a +- sqrt (4a^2 - 12b) / 6
2 sqrt (a^2 - 3b) - 2a > 0
a^2 - 3b > a^2
-3b > 0
b < 0

joyfrancis

for a local maxima at R+, the local minima would also be at some R+.

so the roots of the eqn f'(x) = 0 would both be positive and distinct..so by using sum>0,product>0and D>0 we get 3 conditions which are

a<0

b>0 and

a^2>3b

joyfrancis

Here is the graph for

$y = {x}^{3} - 2{x}^{2} - 5x+5$

this clearly satisfies the condition derived by gaurav of b<0, but here the maxima is at R- so gaurav's answer is incorrect

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