English

Show that the function given by f(x) = 3x + 17 is strictly increasing on R.

Advertisements
Advertisements

Question

Show that the function given by f(x) = 3x + 17 is strictly increasing on R.

Sum
Advertisements

Solution

We have f(x) = 3x + 17

f(x) being a polynomial function, is continuous and derivable on R.

f'(x) `3 > 0, x in R`

⇒ f is strictly increasing on R.

shaalaa.com
  Is there an error in this question or solution?
Chapter 6: Application of Derivatives - Exercise 6.2 [Page 205]

APPEARS IN

NCERT Mathematics Part 1 and 2 [English] Class 12
Chapter 6 Application of Derivatives
Exercise 6.2 | Q 2 | Page 205

RELATED QUESTIONS

Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.


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


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

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

Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.


Prove that the function f(x) = loge x is increasing on (0, ∞) ?


Show that f(x) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?


Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?


Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 9x2 + 12x − 5 ?


Find the interval in which the following function are increasing or decreasing  f(x) = 2x3 − 24x + 107  ?


Find the interval in which the following function are increasing or decreasing f(x) = −2x3 − 9x2 − 12x + 1  ?


Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\] ?


Determine the values of x for which the function f(x) = x2 − 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x2 − 6x + 9 where the normal is parallel to the line y = x + 5 ? 


Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?


The interval of increase of the function f(x) = x − ex + tan (2π/7) is


Every invertible function is


Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)


The consumption expenditure Ec of a person with the income x. is given by Ec = 0.0006x2 + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.


 Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R. 


Find the intervals in which function f given by f(x)  = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .


Test whether the following functions are increasing or decreasing: f(x) = `x-(1)/x`, x ∈ R, x ≠ 0.


Choose the correct option from the given alternatives :

Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.


Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing


Choose the correct alternative:

The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R is


The slope of tangent at any point (a, b) is also called as ______.


Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function


By completing the following activity, find the values of x such that f(x) = 2x3 – 15x2 – 84x – 7 is decreasing function.

Solution: f(x) = 2x3 – 15x2 – 84x – 7

∴ f'(x) = `square`

∴ f'(x) = 6`(square) (square)`

Since f(x) is decreasing function.

∴ f'(x) < 0

Case 1: `(square)` > 0 and (x + 2) < 0

∴ x ∈ `square`

Case 2: `(square)` < 0 and (x + 2) > 0

∴ x ∈ `square`

∴ f(x) is decreasing function if and only if x ∈ `square`


For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.


The function `1/(1 + x^2)` is increasing in the interval ______ 


Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`


The function f(x) = x2 – 2x is increasing in the interval ____________.


The function f (x) = x2, for all real x, is ____________.


In `(0, pi/2),`  the function f (x) = `"x"/"sin x"` is ____________.


The function f: N → N, where

f(n) = `{{:(1/2(n + 1), "If n is sold"),(1/2n, "if n is even"):}` is


Let f(x) = `x/sqrt(a^2 + x^2) - (d - x)/sqrt(b^2 + (d - x)^2), x ∈ R` where a, b and d are non-zero real constants. Then ______.


Read the following passage:

The use of electric vehicles will curb air pollution in the long run.

The use of electric vehicles is increasing every year and the estimated electric vehicles in use at any time t is given by the function V:

V(t) = `1/5 t^3 - 5/2 t^2 + 25t - 2`

where t represents the time and t = 1, 2, 3, ...... corresponds to years 2001, 2002, 2003, ...... respectively.

Based on the above information, answer the following questions:

  1. Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
  2. Prove that the function V(t) is an increasing function. (2)

Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.


Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×