Practicing Success
Given $P(x)=x^4+a x^3+b x^2+c x+d$ such that x = 0 is the only real root of P'(x) = 0. If P(-1) < P(1), then in the interval [-1, 1] |
P(-1) is the minimum and P(1) the maximum of P P(-1) is not minimum but P(1) the maximum of P P(-1) is the minimum but P(1) is not the maximum of P Neither P(-1) is the maximum nor P(1) is the maximum of P |
P(-1) is not minimum but P(1) the maximum of P |
We have, $P(x)=x^4+a x^3+b x^2+c x+d$ $\Rightarrow P'(x)=4 x^3+3 a x^2+2 b x+c$ It is given that x = 0 is a root of P'(x) = 0 ∴ $P'(0)=0 \Rightarrow c=$ Putting c = 0, we get $P'(x)=x\left(4 x^2+3 a x+2 b\right)$ Since x = 0 is the only real root of P'(x) = 0. Therefore, $4 x^2+3 a x+2 b=0$has no real root $\Rightarrow 9 a^2-32 b<0$ It is given that $P(-1)<P(1)$ $\Rightarrow 1-a+b-c+d<1+a+b+c+d$ $\Rightarrow a>0$ But, 9a2 - 32b < 0. Therefore, b > 0 ∴ $P'(x)=x\left(4 x^2+3 a x+2 b\right)>0$ for all $x \in(0,1]$ $\Rightarrow P(x)$ is increasing in $(0,1]$ $\Rightarrow P(1)$ is the maximum value of $P(x)$ Also, $P'(x)=x\left(4 x^2+3 a x+2 b\right)<0$ for all $x \in[-1,0)$ [∵ $4 x^2+3 a x+2 b>0$ for all x] $\Rightarrow P(x)$ is decreasing in $[-1,0)$ $\Rightarrow P(-1)$ is not the minimum value of P. |