Number of integral values of p for which the cubic 2 x3-3 x2+p=0 has 3 real roots (not necessarily distinct), is

Question

Number of integral values of p for which the cubic 2 x3-3 x2+p=0 has 3 real roots (not necessarily distinct), is (A) 1 (B) 2 (C) 3 (D) 4

Tardigrade Admin · Accepted Answer

Draw graph of y=p and y=3 x2-2 x3 Hence, for three roots p ∈[0,1]. Ans. Aliter: Let f(x)=2 x3-3 x2+p f prime(x)=6 x(x-1)

Now, f (0) ⋅ f (1) ≤ 0 ⇒ p( p -1) ≤ 0 ⇒ p ∈[0,1].

Q. Number of integral values of $p$ for which the cubic $2 x^{3} - 3 x^{2} + p = 0$ has 3 real roots (not necessarily distinct), is