Maxima and Minima

Find the equation of a line passing through the point $(4,3)$, which cuts a triangle of minimum area from the first quadrant.

If the function $f\left(x\right)=2x^3-9a{x}^2+12a^{2}x+1$, where $a>0$, attains its maximum and minimum at $p$ and $q$ respectively such that $p^2=q$, then $a$ equals

Find the minimum value of $2 x+3 y$ when $xy=6$.

The stationary points of the curve $y=8x^2-x^4-4$ are____

Find the positive value of '$a$' for which the equality $2\alpha +\beta =8$ holds, where '$\alpha$' and '$\beta$' are the points of maximum and minimum, respectively, of the function $f\left(x\right)=2x^3-9a{x}^2+12a^{2}x+1$

If the curved surface area of right circular cylinder inscribed in a sphere of radius $22 \mathrm{cm}$ is maximum then height of the cylinder will be

If $p$ and $q$ are respectively the global maximum and global minimum of the function $f\left(x\right)=x^2{\mathrm{e}}^{2x}$ on the interval $[-2,2]$, then $p{e}^{-4}+q{e}^4=$

The curve $f\left(x\right)=e^x\sin x$ is defined in the interval $[0,2\pi ].$ The value of $x$ for which the slope of the tangent drawn to the curve at $x$ is maximum, is

The difference between the greatest and least values of the function $f\left(x\right)=-x+\sin 2x$ on $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$ is