k is uniformly chosen from the interval (-5,5) . Let p be the probability that the quartic f(x)=kx^4+(k^2+1)x^2+k has 4 distinct real roots such that one of the roots is less than -4 and the other three are greater than -1. Find the value of 1000p.
k is uniformly chosen from the interval (-5,5) . Let p be the probability that the quartic f(x)=kx^4+(k^2+1)x^2+k has 4 distinct real roots such that one of the roots is less than -4 and the other three are greater than -1. Find the value of 1000p.
let y=x^{2}
Your equation is $\displaystyle ky^2+(k^2+1)y+k$
If you solve this equation for y the value of x will be $\displaystyle \pm y^{1/2}$ So y is greater than or equal to zero if the roots are to be zero.
If the roots are distinct
$\displaystyle y=0$ is not a root and $\displaystyle (k^2+1)^2-4$ is not equal to zero
Find the range of values of k so that one root is below -4 and the others are above -1 and express that range as a proportion of (-5,5) to get a probability.