Application of DerivativeHard
Question
Let p(x) = a0 + a1 x2 + a2x4 + ..... + anx2n be a polynomial ina real variable xwith 0 < a0 < a1 < a1 < .....n The function P(x) has
Options
A.neither a maximum nor a minimum
B.only one maximum
C.only one minimum
D.only one maximum and only one minimum
Solution
Given P(x) = a0 + a1x2 + a2x4 + ...... + anx2n
where, an > an-1 > an-2 ....> a2 > a1 > a0 > 0
⇒ P′(x) = 2a1x + 4a2x3 + .... + 2nanx2n-1
= 2x(a1 + 2a1x2 + .... + nanx2n-2 ..... (i)
where, (a1 + 2a2x2 + 3a3x4 + ..... + nanx2n-2) > 0
for all x ∈ R
Thus,
ie P′(x) changes sign from (-ve) to (+ ve) at x = 0.
∴ P(x) attains minimun at x = 0.
Hence, it has only one minimun at x = 0.
where, an > an-1 > an-2 ....> a2 > a1 > a0 > 0
⇒ P′(x) = 2a1x + 4a2x3 + .... + 2nanx2n-1
= 2x(a1 + 2a1x2 + .... + nanx2n-2 ..... (i)
where, (a1 + 2a2x2 + 3a3x4 + ..... + nanx2n-2) > 0
for all x ∈ R
Thus,
ie P′(x) changes sign from (-ve) to (+ ve) at x = 0.
∴ P(x) attains minimun at x = 0.
Hence, it has only one minimun at x = 0.
Create a free account to view solution
View Solution FreeMore Application of Derivative Questions
The point on the curve y2 = 8x for which the abscissa and ordinate change at the same rate is -...The normal to the curve, x2 + 2xy - 3y2 = 0, at (1, 1)...The angle made by the tangent to the curve x = et cos t, y = et sin t at point t = π/4 with x-axis is -...The normal to the curve x = a (cos θ + θ sin θ) y = a (sin θ - θ cos θ) any point ′&...If a1, a2 ........ an ∈ R+ and a1, a2 .......an = 1 then the least value of (1 + a1 + a21 ) (1 + a2 + a22 )....(1 ...