QJEEM20Ma1e99: JEE Main 2020 Mathematics - Class 12 JEEM20Ma1e99 Solution: If p(x) be a polynomial of degree three that has a local…

Since p(x) has relative extreme at

x = 1 & 2

so p'(x) = 0 at x = 1 & 2

$$\therefore$$ Let p'(x) = A(x – 1) (x – 2)

$$\Rightarrow$$ p(x) = $$\int A\left(x^2-3x+2\right)dx$$

$$\Rightarrow$$ p(x) = $$A\left(\frac{x^2}{3}-3x^2+2x\right)$$ + C ...(1)

As P(1) = 8

From (1)

$$8=A\left(\frac{1}{3}-\frac{3}{2}+2\right)+C$$

$$\Rightarrow$$ 8 = $$\frac{5A}{6}+C$$

$$\Rightarrow$$ 48 = 5A + 5C ...(2)

Also P(2) = 4

From (1)

$$4=A\left(\frac{8}{3}-6+4\right)+C$$

$$\Rightarrow$$ 4 = $$\frac{2A}{3}+C$$

$$\Rightarrow$$ 12 = 2A + 3C ......(3)

Form (3) & (4), C = –12, A = 24

Now p(0) = C = -12