List of top Mathematics Questions on Applications of Derivatives

Let \( f: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \( f''(x)>0 \) for all \( x \in \mathbb{R} \) and \( f'(a-1) = 0 \), where \( a \) is a real number. Let \( g(x) = f(\tan^2 x - 2\tan x + a) \), \( 0<x<\frac{\pi}{2} \).
Consider the following two statements :
(I) \( g \) is increasing in \( (0, \frac{\pi}{4}) \)
(II) \( g \) is decreasing in \( (\frac{\pi}{4}, \frac{\pi}{2}) \)
Then,
If the volume of a solid hemisphere increases at a uniform rate, prove that its surface area varies inversely as its radius.
Find the absolute maximum value of $f(x)=\cos x+\sin^2 x$, where $x\in[0,\pi]$.
Find the equation of the tangent to the curve \(y = x^2\) at the point (1,1).
A boat takes 4 hours to travel 40 km downstream and 6 hours to return. The speed of the stream is:
The equation of the tangent to the curve \(y=x^3-2x+1\) at \((1,0)\) is:
A person travels from A to B at 40 km/h and returns at 60 km/h. The average speed for the entire journey is:
Given that \( f(x) = \frac{\log x}{x} \), find the point of local maximum of \( f(x) \).
A wire of length 20 units is divided into two parts such that the product of one part and the cube of the other part is maximum. The product of these parts is:

If \( x = a(0 - \sin \theta) \), \( y = a(1 + \cos \theta) \), find \[ \frac{dy}{dx}. \] 

Find the least value of ‘a’ for which the function \( f(x) = x^2 + ax + 1 \) is increasing on the interval \( [1, 2] \).

Find the interval in which the function \[ f(x) = 4x^3 - 6x^2 - 72x + 30 \] is (i) increasing and (ii) decreasing.
Find the equation of tangent at $t = \dfrac{\pi}{2}$ on the curve $x = a \sin^{3}t$, $y = b \cos^{3}t$.
Prove that the function \[ f(x) = \tan^{-1} (\sin x + \cos x), x > 0 \text{ is always increasing function on } (0, \frac{\pi}{4}). \]
Find the equation of tangent at the point \( (am^2, am^3) \) on the curve \[ ay^2 = x^3. \]
If the length of a rectangle is decreasing at the rate of 3 cm/min and width is increasing at the rate of 2 cm/min, then find the rate of change in perimeter of the rectangle when \( x = 10 \) cm and \( y = 6 \) cm, where \( x \) = length and \( y \) = width.
Find those points on the curve \[ \frac{x^2}{4} + \frac{y^2}{25} = 1, \] on which the normal is parallel to the x-axis.
Find the intervals in which the function $f(x)=\sin x + \cos x,\; 0\leq x \leq 2\pi$ is increasing or decreasing.
Find the point on the curve \[ y = x^3 - 11x + 5 \] at which the tangent is \[ y = x - 11. \]
Find the equation of tangent of the curve $y = \cos(x+y)$, $-2\pi \leq x \leq 2\pi$, which is parallel to the line $x + 2y = 0$.
The slope of the normal to the curve $y = 2x^{2} + 3\sin x$ at $x = 0$ will be:
Ahousing society wants to commission a swimming pool for its residents. For this, they have to purchase a square piece of land and dig this to such a depth that its capacity is 250 cubic metres. Cost of land is 500 per square metre. The cost of digging increases with the depth and cost for the whole pool is 4000 (depth)2. Suppose the side of the square plot is x metres and depth is h metres.
On the basis of the above information, answer the following questions
Given:- Square plot side = x metres
- Depth = h metres
- Volume = x2h = 250 cubic metres
- Cost of land: 500 per sq.metre
- Cost of digging: 4000 ×h

In an agricultural institute, scientists conduct experiments with varieties of seeds to grow them in different environments for producing healthy plants and obtaining higher yields.
A scientist observed that a particular seed grew very fast after germination. He recorded the growth of the plant from the time of germination and modeled its growth with the function:
Given:
\( f(x) = \frac{1}{3}x^3 - 4x^2 + 15x + 2 \),   \( 0 \leq x \leq 10 \)
where \( x \) is the number of days the plant is exposed to sunlight.
On the basis of the above information, answer the following questions: 

Find the intervals in which the function
\( f(x) = x^4 - 4x^3 + 4x^2 + 15 \) is increasing .
A cylindrical gas container is closed at the top and open at the bottom. If the iron plate of the top is (5)/(4) times as thick as the plate forming the cylindrical sides, find the ratio of the radius to the height of the cylinder using minimum material for the same capacity.