Given $$P\left( x \right) = {x^4} + a{x^3} + b{x^2} + cx + d$$ such that $$x=0$$ is the only
real root of $$P'\,\left( x \right) = 0.$$ If $$P\left( { - 1} \right) < P\left( 1 \right),$$ then in the interval $$\left[ { - 1,1} \right]:$$
A
$$P(-1)$$ is not minimum but $$P(1)$$ is the maximum of $$P$$
B
$$P(-1)$$ is the minimum but $$P(1)$$ is not the maximum of $$P$$
C
Neither $$P(-1)$$ is the minimum nor $$P(1)$$ is the maximum of $$P$$
D
$$P(-1)$$ is the minimum and $$P(1)$$ is the maximum of $$P$$
Explanation
We have $$P\left( x \right) = {x^4} + a{x^3} + b{x^2} + cx + d$$
Max $$P\left( x \right) = P\left( 1 \right)$$ and Min $$P\left( x \right) = P\left( 0 \right)$$
$$ \Rightarrow P\left( { - 1} \right)$$ is not minimum but $$P\left( 1 \right)$$ is the maximum of $$P.$$
2
AIEEE 2009
MCQ (Single Correct Answer)
Let $$f\left( x \right) = x\left| x \right|$$ and $$g\left( x \right) = \sin x.$$
Statement-1: gof is differentiable at $$x=0$$ and its derivative is continuous at that point.
Statement-2: gof is twice differentiable at $$x=0$$.
A
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
B
Statement-1 is true, Statement-2 is false
C
Statement-1 is false, Statement-2 is true
D
Statement-1 is true, Statement-2 is true Statement-2 is a correct explanation for Statement-1
Explanation
Given that $$f\left( x \right) = x\left| x \right|\,\,$$ and $$\,\,g\left( x \right) = \sin x$$
So that go
$$f\left( x \right) = g\left( {f\left( x \right)} \right)$$
$$ = g\left( {x\left| x \right|} \right) = \sin x\left| x \right|$$