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

If $f(x) = \frac{\sin(x + a)}{\sin(x + b)}$,$a \neq b$,then $f$ is......

${a_1, a_2, ....., a_n, .....}$ is a progression where $a_n = \frac{n^2}{n^3 + 200}$. The largest term of this progression is

Let $f(x) = 1 + 2x^2 + 2^2x^4 + \dots + 2^{10}x^{20}$. Then $f(x)$ has:

Let a function $f(x)$ be continuous in an interval $[a, b]$. Let $\delta > 0$ be a very small real number. Let $c \in (a, b)$ be such that $f(c - \delta) < f(c)$ and $f(c + \delta) < f(c)$ for every $\delta > 0$. Let $(f(\alpha - \delta) - f(\alpha))(f(\alpha + \delta) - f(\alpha)) < 0$ for all $\alpha \in (a, b)$ and $\alpha \neq c$. Then:

The minimum value of the slope of the tangent to the curve $y=x^3-3x^2+2x+93$ is