List of top Mathematics Questions on Applications of Derivatives asked in VITEEE

The value of \( x \) in the interval \( [4, 9] \) at which the function \[ f(x) = \sqrt{x} \] \text{satisfies the mean value theorem is:}

If \( f(x) = x^3 + bx^2 + cx + d \) and \( 0<b^2<c \), then in \( (-\infty, \infty) \),

A wire 34 cm long is to be bent in the form of a quadrilateral of which each angle is 90°. What is the maximum area which can be enclosed inside the quadrilateral?

The radius of a right circular cylinder increases at the rate of 0.1 cm/min, and the height decreases at the rate of 0.2 cm/min. The rate of change of the volume of the cylinder, in cm\(^3\)/min, when the radius is 2 cm and the height is 3 cm is:

If at \( x = 1 \), the function \( x^4 - 62x^2 + ax + 9 \) attains its maximum value on the interval \( [0, 2] \), then the value of \( a \) is:

The value of \( c \) in Rolle's Theorem for the function \[ f(x) = e^x \sin x, \, x \in [0, \pi] \] is:

A football is inflated by pumping air in it. When it acquires spherical shape its radius increases at the rate of \( 0.02 \, \text{cm/s} \). The rate of increase of its volume when the radius is \( 10 \, \text{cm} \) is:

The volume \( V \) and depth \( x \) of water in a vessel are connected by the relation \[ V = 5x - \frac{x^2}{6} \] and the volume of water is increasing, at the rate of 5 cm\(^3\)/sec, when \( x = 2 \) cm. The rate at which the depth of water is increasing is

The difference between greatest and least value of \[ f(x) = 2 \sin x + \sin 2x, \, x \in \left[ 0, \frac{3\pi}{2} \right] \] is

The function \( f(x) = \sin x - kx - c \), where \( k \) and \( c \) are constants, decreases always when

The distance moved by the particle in time \( t \) is given by \[ s = t^3 - 12t^2 + 6t + 8 \] At the instant, when its acceleration is zero, its velocity is

The distance moved by the particle in time \( t \) is given by \[ s = t^3 - 12t^2 + 6t + 8 \] At the instant, when its acceleration is zero, its velocity is

Let \( f(x) = 25x^{24}(1 - x)^{75} \), on the interval \( [0, 1] \), then \[ f'(x) = 25x^{24}(1 - x)^{75} - 75x^{25}(1 - x)^{74} = 25x^{24}(1 - x)^{74}[(1 - x) - 3x] \]

Let \( f'(x) \) be differentiable for all \( x \). If \( f(1) = -2 \) and \[ f'(x) \geq 2 \quad \forall x \in [1, 6], \] then:

The equation of normal to the curve \( y = (1 + x) + \sin^{-1}(\sin x) \) at \( x = 0 \) is?

The value of from the Lagrange's mean value theorem for which \( f(x) = \sqrt{25 - x^2} \) in the interval \( [1,5] \) is?

The minimum value of \( \frac{x}{\log x} \) is?

The triangle formed by the tangent to the curve \( y = x^2 + x + 1 \) at the point \( (1, 3) \) and the coordinate axes lies in the first quadrant. If its area is 2, then the value of \( b \) is?

For which of the following values of \( m \), the area of the region bounded by the curve \( y = x - x^2 \) and the line \( y = mx \) equals 5?

The rate of change of the surface area of a sphere of radius \( r \), when the radius is increasing at the rate of \( 2 \, \text{cm/s} \), is proportional to

The maximum value of \( 4 \sin^2 x - 12 \sin x + 7 \) is:

The two curves \(y=3^x\) and \(y=5^x\) intersect at an angle