The minimum value of $2x^2+x-1$ is:

The sufficient conditions for the function $f:R \to R$ to have a local maximum at $x = a$ are:

If $P$ is a point on the segment $AB$ of length $12 \text{ cm}$,then the position of $P$ for $AP^{2} + BP^{2}$ to be minimum is such that

For the function $f(x) = x^4(12\ln x - 7)$,match the following columns:

Column-$I$	Column-$II$
$(A)$ If $(a, b)$ is a point of inflection,then $a - b$ equals	$(P)$ $3$
$(B)$ If $e^t$ is the point of local minimum,then $12t$ equals	$(Q)$ $1$
$(C)$ If the graph is concave down on $(d, e)$,then $d + 3e$ equals	$(R)$ $4$
$(D)$ If the graph is concave up on $(p, \infty)$,then $p$ equals	$(S)$ $8$

Let $P(x)$ be a polynomial of degree $3$ having an extreme value at $x=1$. If $\lim _{x \rightarrow 0}\left(\frac{P(x)+4}{x^2}+2\right)=6$,then $\left(\frac{d P}{d x}\right)_{x=\frac{1}{2}}=$

$A$ curve with equation of the form $y = ax^4 + bx^3 + cx + d$ has zero gradient at the point $(0, 1)$ and also touches the $x$-axis at the point $(-1, 0)$. Then the values of $x$ for which the curve has a negative gradient are: