Application of Derivatives | Mathematics | NDA Previous Year Questions

Consider the following statements

1. f(x) = ln x is an increasing function on (0, $$\infty$$).

2. f(x) = e^x $$-$$ x(ln x) is an increasing function on (1, $$\infty$$).

Which of the above statements is/are correct?

At what value of x does the function attain maximum value?

The maximum value of the function is

The function f(x) is an increasing function in the interval

The function f(x) is a decreasing function in the interval

Which one of the following statements is correct in respect of the function f(x) = x³ sin x?

Which of the following statements is/are correct?

I. f(x) is increasing in the interval {$$-$$1, 2}.

II. f(x) is decreasing in the interval {2, 3}.

Select the correct answer using the code given below.

Which of the following statements are correct?

I. f(x) is continuous at x = 2.

II. f(x) attains greatest value at x = 2.

III. f(x) is differentiable at x = 2.

Select the correct answer using the code given below.

Let $$f(x) = \left\{ {\matrix{ {{{{e^x} - 1} \over x},} & {x > 0} \cr {0,} & {x = 0} \cr } } \right.$$ be a real valued function. Which one of the following statements is correct?

Which one of the following statements is correct?

NDA 2016 Paper 1

Which one of the following statements is correct?

NDA 2016 Paper 1

Which one of the following is correct in respect of the function

$$f(x) = x(x - 1)(x + 1)$$ ?

NDA 2017 Paper 2

The maximum value of $${{\ln x} \over x}$$ is

NDA 2017 Paper 2

A cylindrical jar without a lid has to be constructed using a given surface area of a metal sheet. If the capacity of the jar is to be maximum, then the diameter of the jar must be k times the height of the jar. The value of k is

NDA 2017 Paper 2

What is the length of the longest interval in which the function $$f(x) = 3\sin x - 4{\sin ^3}x$$ is increasing?

NDA 2017 Paper 1

Let $$f(x) = x + {1 \over x}$$, where x$$\in$$ (0, 1). Then which one of the following is correct?

NDA 2017 Paper 1

Let the slope of the curve y = cos^$$-$$1 (sin x) be tan$$\theta$$. Then, the value of $$\theta$$ in the interval (0, $$\pi$$) is

Which one of the following is correct in respect of the function $$f(x) = x\sin x + \cos x + {1 \over 2}{\cos ^2}x$$ ?

In which one of the following intervals is the function $$f(x) = {x^2} - 5x + 6$$ decreasing?

A flower-bed in the form of a sector has been fenced by a wire of 40 m length. If the flower-bed has the greatest possible area, then what is the radius of the sector?

What is the minimum value of $${[x(x - 1) + 1]^{{1 \over 3}}}$$, where 0 $$\le$$ x $$\le$$ 1 ?

If the area of the trapezium is the maximum possible, then what is $$\alpha$$ equal to?

NDA 2018 Paper 1

If the area of the trapezium is maximum, what is the length of the fourth side?

NDA 2018 Paper 1

What is the maximum area of the trapezium?

NDA 2018 Paper 1

What is the slope of the curve at the point of intersection P?

How much angle does the tangent at P make with y-axis?

What is the equation of tangent to the curve at P?

If $$f(x) = {{{x^3}} \over 3} - {{5{x^2}} \over 2} + 6x + 7$$ increases in the interval T and decreases in the interval S, then which one of the following is correct?

In which one of the following intervals is the function increasing?

In which one of the following intervals is the function decreasing?

A given quantity of metal is to be case into a half cylinder (i.e. with a rectangular base and semicircular ends). If the total surface area is to be minimum, then the ratio of the height of the half cylinder to the diameter of the semicircular ends is

NDA 2019 Paper 1

The minimum distance from the point (4, 2) to y² = 8x is equal to

NDA 2019 Paper 1

What is the minimum value of a²x + b²y where xy = c² ?

NDA 2019 Paper 1

What is the greatest value of f(x)?

NDA Mathematics 1st September 2024

What is the least value of f(x)?

NDA Mathematics 1st September 2024

The non-negative values of $b$ for which the function $\frac{16x^3}{3} - 4bx^2 + x$ has neither maximum nor minimum in the range $x>0$ is

What is $\varphi(a)$ equal to?

What is $\varphi'(a)$ equal to?

Which of the following is/are correct?

1. $f'(0) = 0$

2. $f''(0) < 0$

Select the correct answer using the code given below:

The function $y$ has a relative maxima at $x = 0$ for

What is the maximum value of y?

NDA Mathematics 3 September 2023

What is the maximum value of xy ?

NDA Mathematics 3 September 2023

Consider the following statements :

1. f(x) = In x is increasing in (0, ∞)

2. $g(x)=e^x+e^{\frac{1}{x}} $ is decreasing in (0, ∞)

Which of the statements given above is/are correct ?

NDA Mathematics 16 April 2023

What is the derivative of sin² x with respect to cos²x ?

NDA Mathematics 16 April 2023

Under which one of the following conditions does the function f(x) = (p sec x)2 + (q cosec x)2 attain minimum value ?

NDA Mathematics 4 September 2022

Where does the function f(x) = $\displaystyle\sum_{j=1}^7$(x − j)2 attain its minimum value ?

NDA Mathematics 4 September 2022

How many extreme values does sin4x + 2x, where $0 < x < \frac{\pi}{2} $ have ?

NDA Mathematics 10 April 2022

If the derivative of the function$f(x) =\frac{m}{x} +2nx + 1$ vanishes at x = 2, then what is the value of m + 8n ?

NDA Mathematics 10 April 2022

If xy = 4225 where x, y are natural numbers, then what is the minimum value of x + y ?

NDA Mathematics 10 April 2022

Consider the following statements in respect of the function f(x) = x² + 1 in the interval [1, 2]:

1. The maximum value of the function is 5.

2. The minimum value of the function is 2.

Which of the above statements is/are correct?

Where does the tangent to the curve y = e^x at the point (0, 1) meet x-axis?

Consider the following statements in respect of the function f(x) = x + $\rm \frac{1}{x}$:

1. The local maximum value of f(x) is less than its local minimum value.

2. The local maximum value of f(x) occurs at x = 1.

Which of the above statements is/are correct?

What is the maximum area of a rectangle that can be inscribed in a circle of radius 2 units?

What is the condition that f(x) = x³ + x² + kx has no local extremum?

If the function f(x) = x² - kx is monotonically increasing in the interval (1, ∞), then which one of the following is correct?

The tangent to the curve x² = y at (1, 1) makes an angle θ with the positive direction of x-axis. Which one of the following is correct?

Consider the following statements in respect of the function f(x) = sin x:

1. f(x) increases in the interval (0, π).

2. f(x) decreases in the interval $\left(\dfrac{5\pi}{2},3\pi\right).$

Which of the above statements is/are correct?

The curve y = -x³ + 3x² + 2x - 27 has the maximum slope at:

If x + y = 20 and P = xy, then what is the maximum value of P?

What is the maximum value of sin 2x ⋅ cos 2x?