Solveeit Logo
Login

Question

Question: If all roots of equation \(x^{3} - 3x + k = 0\) are real, then range of value of k...

If all roots of equation x3−3x+k=0x^{3} - 3x + k = 0 are real, then range of value of k

A

(–2, 2)

B

[–2, 2]

C

Both

D

None of these

Answer

(–2, 2)

Explanation

Solution

Let f(x)=x3−3x+kf(x) = x^{3} - 3x + k, then f(x)=3x2−3f(x) = 3x^{2} - 3 and so

f(x)=0⇒x=±1f(x) = 0 \Rightarrow x = \pm 1. The values of f(x) at x=−∞,−1,1,∞x = - \infty, - 1,1,\infty are :

x: & - \infty & - 1 & 1 & \infty \\ f(x): & - \infty & k + 2 & k - 2 & \infty \end{matrix}$$ If all roots of given equation are real, then $k + 2 > 0$ and $k - 2 < 0 \Rightarrow - 2 < k < 2$. Hence the range of *k* is (–2, 2)