Commerce (English Medium) इयत्ता १२ - CBSE Question Bank Solutions

Find the values of x for  `y = [x(x - 2)]^2` is an increasing function.

Prove that  y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`

Prove that the logarithmic function is strictly increasing on (0, ∞).

Prove that the function f given by f(x) = x² − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).

Which of the following functions are strictly decreasing on `(0, pi/2)`?

1. cos x
2. cos 2x
3. cos 3x
4. tan x

On which of the following intervals is the function f given byf(x) = x¹⁰⁰ + sin x –1 strictly decreasing?

Find the least value of a such that the function f given by f (x) = x² + ax + 1 is strictly increasing on [1, 2].

Let I be any interval disjoint from (−1, 1). Prove that the function f given by `f(x) = x + 1/x` is strictly increasing on I.

Prove that the function f given by f(x) = log sin x is strictly increasing on `(0, pi/2)` and strictly decreasing on `(pi/2, pi)`

Prove that the function f given by f(x) = log cos x is strictly decreasing on `(0, pi/2)` and strictly increasing on `((3pi)/2, 2pi).`

Prove that the function given by f (x) = x³ – 3x² + 3x – 100 is increasing in R.

The interval in which y = x² e^–x is increasing is ______.

Find the intervals in which the function f given by `f(x) = (4sin x - 2x - x cos x)/(2 + cos x)` is (i) increasing (ii) decreasing.

Find the intervals in which the function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.

Let f be a function defined on [a, b] such that f '(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).

Find an anti derivative (or integral) of the following function by the method of inspection.

sin 2x

Find an anti derivative (or integral) of the following function by the method of inspection.

Cos 3x

Find an anti derivative (or integral) of the following function by the method of inspection.

e^2x

Find an anti derivative (or integral) of the following function by the method of inspection.

(axe + b)²

Find an antiderivative (or integral) of the following function by the method of inspection.

sin 2x – 4 e^3x