The maximum value of $a$ such that the second derivative of $x^4+ax^3+\frac{3x^2}{2}+1$ is positive for all real $x$ is

If $y = \sin x \sin 3x$,then find the $n^{th}$ derivative $y_n$.

If $x^2+xy+y^2=k$,then $\frac{d^2y}{dx^2}=$

If $f(x) = \sin x \cdot \sin 2x \cdot \sin 3x$ and $f''(x) = a(\sin bx) + c(\sin dx) + e(\sin kx)$,then the value of $(a+c+e) - (b+d+k)$ equals

If $x^x y^y=e^e$,then $\left(\frac{d^2 y}{d x^2}\right)_{(e, e)}=$

$A$ polynomial $P(x)$ with real coefficients has the property that $P^{\prime \prime}(x) \neq 0$ for all $x$. Suppose $P(0) = 1$ and $P^{\prime}(0) = -1$. What can you say about $P(1)$?