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Monday, January 12, 2015

2015/006) Find the maximum value of P(x)=x^3 −3x on the set of all real x that satisfies the inequality x^4 −13x^2 +36\le0 .

we have x^4-13x^2+ 36 \le 0
or (x^2-9)(x^2-4) \le 0
or 4\le x^2\le 9

so we get 2 ranges for x

1) -3\le x\le -2
2)  2\le x\le 3

we need to find the maximum of x^3-3x in this ranges

x^3-3x = x(x^2-3)

in the 1st range x is negative  and x^2-3 is positive so in the range x^3-3x is -ve

for x\gt \sqrt{3} and hence for x \ge 2 both x and x^2- 3 are positive and increasing so the highest value is at highest x that is x = 3 so the value of x^3-3x=18

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