Question

let p(x)=x^4+ax^3+bx^2+cx+d, where a, b, c and d are constants. if p(1) = 10, p(2) = 20 and p(3) = 30 .compute (p(12)+p(-8))÷10

Answer 1

 

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Given P(x) = x4 +ax3 + bx2 +cx + d such that...

Given P(x) = x4 +ax3 + bx2 +cx + d such that x = 0 is the only real root of P' (x) =0. If P(–1) < P(1), then in the interval [–1,1]

5 years ago

Answers : (1)

Given that P(x) = x4 +ax3 + bx2 + cx + d

Hence, P’(x) = 4x3 + 3ax2 + 2bx + c

Now as P’(0) = 0 so the above equation gives c = 0.

Hence, P’(x) = x (4x2 + 3ax + 2b)

As x = 0 is teh only real root of P’(x) = 0, roots of 4x2 + 3ax + 2b must have imaginary roots.

Hence, 4x2 + 3ax + 2b > 0  x ∈ R.

Thus P’(x) < 0 for x < 0

and P’(x) > 0 for x > 0

This gives that x = 0 is a point of local minima.

As P(-1) < P(1), we get P(1) is maximum but P(-1) is not minimum of P on [-1, 1].

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