10 Question MCQ (including Assertion)

Assertion (A): The function \(f(x)=\sin (x)\) is increasing on the interval \(\left(0, \frac{\pi}{2}\right)\).
Reason (R): The derivative, \(f^{\prime}(x)=\cos (x)\), is positive for all \(x\) in the interval ( \(0, \frac{\pi}{2}\))

For a function \(f(x)\) on a closed interval \([a, b]\), the absolute maximum value can occur:

If \(f^{\prime}(x)\) changes sign from negative to positive as \(x\) increases through a point \(c\), then \(c\) is \(a\) :

The rate of change of a circle's circumference with respect to its radius is:

The "point of diminishing returns" in economics, where adding more input starts to yield smaller increases in output, corresponds mathematically to:

The height of a projectile is given by \(h(t)=-16 t^2+v_0 t+h_0\). The maximum height is reached when:

The radius of a spherical balloon is increasing. The rate of change of its volume is fastest when:

If a function \(f(x)\) is decreasing on \((0,2)\) and increasing on \((2,5)\), then at \(x=2\) the function has a :

The interval on which the function \(f(x)=2 x^3+9 x^2+12 x-1\) is decreasing is: