Applications of Derivatives

Let f be a twice differentiable function on (1,6). If f(2) = 8, f'(2) = 5, f'(x) ≥ 1 and f''(x) ≥ 4, for all x ∈ (1,6), then:

If Lim(x→0) [(tan x)/x]^(1/x²) = p, then 96log? p is equal to

The function, f(x) = (3x – 7)x²/³, x ∈ R, is increasing for all x lying in:

If $\mathrm{c}$ $$ is a point at which Rolle's theorem holds for the function, ${}_{}\left(\frac{{}^{}}{}\right)$ $f\left(x\right)={\mathrm{l}\mathrm{o}\mathrm{g}}_e?\left(\frac{x^2+\alpha }{7x}\right)$ in the interval $\left[\mathrm{3,4}\right]$ $$ , where $$ , $\alpha \in R$ then ${}^{}$ $f^{\mathrm{\text{'}}\mathrm{\text{'}}}\left(c\right)$ is equal to:

If the lines x+y=a and x-y=b touch the curve y=x²-3x+2 at the points where the curve intersects the x-axis, then a/b is equal to

Which of the following point lies on the tangent to the curve x⁴eʸ + 2√(y+1) = 3 at the point (1,0)?