Advertisements
Advertisements
प्रश्न
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
Advertisements
उत्तर
\[f\left( x \right) = x^3 - 3 x^2 + 4x\]
\[f'\left( x \right) = 3 x^2 - 6x + 4\]
\[ = 3\left( x^2 - 2x \right) + 4\]
\[ = 3\left( x^2 - 2x + 1 \right) - 3 + 4\]
\[ = 3 \left( x - 1 \right)^2 + 1 > 0, \forall x \in R\]
\[\text { Hence , f(x) is strictly increasing on R } .\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the function f given by f(x) = 2x3 − 3x2 − 36x + 7 is
- Strictly increasing
- Strictly decreasing
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
The interval in which y = x2 e–x is increasing is ______.
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) = x4 − 4x ?
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 − 6x + 3 is increasing on the interval [4, 6] ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
The function f(x) = xx decreases on the interval
Find the intervals in which the function \[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] is
(a) strictly increasing
(b) strictly decreasing
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
Find the values of x for which the following func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the values of x for which Revenue is increasing.
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing function
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
If f(x) = x3 – 15x2 + 84x – 17, then ______.
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
Which of the following functions is decreasing on `(0, pi/2)`?
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
The function f (x) = 2 – 3 x is ____________.
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
The function f(x) = x3 + 6x2 + (9 + 2k)x + 1 is strictly increasing for all x, if ____________.
Let x0 be a point in the domain of definition of a real valued function `f` and there exists an open interval I = (x0 – h, ro + h) containing x0. Then which of the following statement is/ are true for the above statement.
The function f(x) = xex(1 − x), x ∈ R, is ______.
For \[f(x)=4x^3-6x^2-72x+30\], which factorization of \[f'(x)\] is correct?
The critical points \[x=-2\] and \[x=3\] divide the real line into which three intervals?
For \[f'(x)=12(x-3)(x+2)\], what is the nature of \[f\] on \[(-2,3)\]?
