Application of ...

At what values of $$a$$, the curve $$x^4+3ax^3+6x^2+5$$ is not situated below any of its tangent lines

The minimum value of the polynomial.
$$p(x)=3{ x }^{ 2 }-5x+2$$

If the function $$\displaystyle f\left( x \right)=\left( { a }^{ 2 }-3a+2 \right) \cos { \frac { x }{ 2 }  } +\left( a-1 \right) x$$ possesses critical points, then $$a$$ belongs to the interval

The curve which passes through $$(1, 2)$$ and whose tangent at any point has a slope that is half of slope of the line joining origin to the point of contact, is -

The lines tangent to the curve $$x^3-y^3+x^2y-yx^2+3x-2y=0$$ and $$x^5-y^4+2x+3y=0$$ at the origin intersect at an angle $$\theta$$ equal to

A curve $$\displaystyle y=f\left( x \right) ;\left( y>0 \right) $$  passes thorugh $$(1,1)$$ and at point $$\displaystyle P(x,y)$$ tangents cuts x-axis and y-axis at A and B respectively. If P divides AB  internally in the ratio $$3 : 2$$, then the value of $$\displaystyle f\left( \frac { 1 }{ 8 }  \right) $$ is

If the line $$x -y = 0$$ is tangent to $$f(x) = b \ln x - x$$, then $$b$$ lies in the interval

If f(x) = $$\dfrac{x}{ sin x}$$ and g(x) = $$\dfrac{x}{tanx}$$ where 0<x $$\leq$$ 1, then in this interval $$f(x)$$ is

If the tangent to the curve $$x = a(8 + sin \theta), y = a(1 + cos \theta )$$ at $$\theta = \displaystyle \frac{\pi}{3}$$ makes an angle $$\alpha$$ with x-axis, then $$\alpha$$ is equal to

Determine the intervals over which the function is decreasing, increasing, and constant.