The minimum value of the function $f(x) = \frac{40}{3x^4 + 8x^3 - 18x^2 + 60}$ is:

The denominator of a fraction is $16$ more than the square of the numerator. Find the least value of the fraction.

If $f(x)=\sin x+2 \cos ^{2} x$ for $\frac{\pi}{4} \leq x \leq \frac{3 \pi}{4}$,then $f$ attains its

$A$ stone thrown upwards has its equation of motion $s = 490t - 4.9t^2$. Then the maximum height reached by it is:

Let $\alpha = \sum_{k=1}^{\infty} \sin^{2k}\left(\frac{\pi}{6}\right)$. Let $g:[0,1] \rightarrow \mathbb{R}$ be the function defined by $g(x) = 2^{\alpha x} + 2^{\alpha(1-x)}$. Then,which of the following statements is/are $TRUE$?
$(A)$ The minimum value of $g(x)$ is $2^{7/6}$
$(B)$ The maximum value of $g(x)$ is $1 + 2^{1/3}$
$(C)$ The function $g(x)$ attains its maximum at more than one point
$(D)$ The function $g(x)$ attains its minimum at more than one point

If $x$ and $y$ are two positive numbers such that $x+y=32$,then the minimum value of $x^2+y^2$ is,