CUET Mathematics: CalculusApplication of Derivatives. Free, no login required.

$f(x)=\sin x+\frac{1}{2} \cos 2 x \text { in }\left[0, \frac{\pi}{2}\right]$

(A) $f^{\prime}(x)=\cos x-\sin 2 x$

(B) The critical points of the function are $x=\frac{\pi}{6}$ and $x=\frac{\pi}{2}$

(C) The minimum value of the function is $2$

(D) The maximum value of the function is $\frac{3}{4}$

Choose the correct answer from the options given below :

If $f(x)=2\left(\tan ^{-1}\left(e^{x}\right)-\frac{\pi}{4}\right)$, then $f(x)$ is :

If the lengths of the three sides of a trapezium other than the base are $10 \mathrm{~cm}$ each, then the maximum area of the trapezium is:

The equation of the tangent to the curve $\mathrm{x}^{\frac{5}{2}}+\mathrm{y}^{\frac{5}{2}}=33$ at the point $(1,4)$ is :

The critical points of $f(x) = x^3 + x^2 + x + 1$ are

The volume of a cube is increasing at the rate of 27 cm$^3$/s. How fast is the surface area increasing when the length of the cube is 12 cm.

The intervals for which $f(x) = x^4 - 2x^2$ is increasing are :

The slope of the normal to the curve $y = 2x^2 - 4$ at P (1, -2) is :

The minimum value of $f(x) = |2x - 1|$ is

Interval in which the function $f(x) = 2x^3 - 3x^2 - 12x + 10$ is decreasing is :

The angle of intersection between the curves $y = 4 - x^2$ and $y = x^2$ is :

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

The equation of tangent to the curve $x = a \cos^3 t, y = a \sin^3 t$ at t is :

If the rate of change of area of a circle is equal to the rate of change of its diameter, then its radius is equal to :

The appropriate change in the volume V of a cube of side $x$ metres caused by increasing the side by 2% is :

The maximum value of $2x^3 - 24x + 107$ in the interval $[1, 3]$ is :

The slope of the tangent to the curve $x = at^2$, $y = 2at$ at 't' is :

If the function $f(x) = x^4 - 62x^2 + ax + 9$ attains its local maximum value at $x = 1$, then a is equal to :

The interval in which the function $f(x) = 2x^3 - 3x^2 - 36x + 7$ is strictly decreasing is :

The two curves $x^3 - 3xy^2 + 15 = 0$ and $3x^2 y - y^3 + 17 = 0$ :

The equation of tangent to the curve given by $x = a\sin^3 t$, $y = b\cos^3 t$ at a point where $t = \frac{\pi}{2}$ is :

The rate of change in area of a triangle having sides 10 cm and 12 cm when the variable angle between them is $\theta = 60°$, is :

The interval in which the function $f(x) = 10 - 6x - 2x^2$ is decreasing is :

The value of C which satisfies Rolle's Theorem for $f(x) = \sin^4 x + \cos^4 x$ in $\left[0, \frac{\pi}{2}\right]$. Then C is :

Angle between tangents to the curve $y = x^2 - 5x + 6$ at the points (2, 0) and (3, 0) is :

The rate of change of the area of a circular disc with respect to its circumference when radius is 3 is :

The interval in which the $f(x) = \sin x - \cos x$, $0 \leq x \leq 2\pi$ is strictly decreasing is :

If $2x + y = 6$ then the maximum value of $x^2 y$ is :

If the function $f(x) = x^2 - ax - 2$ is strictly decreasing on $(2, 3)$ then $a$ lies in the interval.