The under of integers $n$ for which $3 x^{3}-25 x+n=0$ has three real roots is

• A

  1
• B

  25
• Correct answer
  C

  55
• D

  Infinite

Solution

Let $f(x)=3 x^{3}-25 x$ $f^{\prime}(x)=9 x^{2}-25$ $f^{\prime}(x)=0 \Rightarrow \pm \frac{5}{3}$ $f(-5 / 3)=250 / 9 \simeq 27.7$ $f(5 / 3)=-250 / 9 \simeq-27.7$ $\therefore$ n can take 55 values

Hence, the correct answer is option: 'C'.