Advertisements
Advertisements
प्रश्न
Find the interval in which the following function are increasing or decreasing f(x) = 6 − 9x − x2 ?
Advertisements
उत्तर
\[\text { When }\left( x - a \right)\left( x - b \right)>0 \text { with} a < b, x < a \ or \ x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = 6 - 9x - x^2 \]
\[f'\left( x \right) = - 2x - 9\]
\[\text { For }f(x) \text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow - 2x - 9 > 0\]
\[ \Rightarrow - 2x > 9\]
\[ \Rightarrow x < \frac{- 9}{2}\]
\[ \Rightarrow x \in \left( - \infty , \frac{- 9}{2} \right)\]
\[\text { So,}f(x)\text { is increasing on } \left( - \infty , \frac{- 9}{2} \right) . \]
\[\text { For }f(x) \text { to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow - 2x - 9 < 0\]
\[ \Rightarrow - 2x < 9\]
\[ \Rightarrow x > \frac{- 9}{2}\]
\[ \Rightarrow x \in \left( \frac{- 9}{2}, \infty \right)\]
\[\text { So,}f(x)\text { is decreasing on }\left( \frac{- 9}{2}, \infty \right).\]
APPEARS IN
संबंधित प्रश्न
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/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.
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).
Prove that the function f(x) = loge x is increasing on (0, ∞) ?
Find the interval in which the following function are increasing or decreasing f(x) = x2 + 2x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = x8 + 6x2 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?
Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?
Let f(x) = x3 − 6x2 + 15x + 3. Then,
Function f(x) = | x | − | x − 1 | is monotonically increasing when
Function f(x) = ax is increasing on R, if
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
Test whether the following functions are increasing or decreasing: f(x) = `x-(1)/x`, x ∈ R, x ≠ 0.
Find the values of x for which the following functions are strictly increasing:
f(x) = 3 + 3x – 3x2 + x3
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
The price P for the demand D is given as P = 183 + 120D − 3D2, then the value of D for which price is increasing, is ______.
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`
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
The function f(x) = tanx – x ______.
2x3 - 6x + 5 is an increasing function, if ____________.
The function f(x) = x3 + 6x2 + (9 + 2k)x + 1 is strictly increasing for all x, if ____________.
State whether the following statement is true or false.
If f'(x) > 0 for all x ∈ (a, b) then f(x) is decreasing function in the interval (a, b).
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
Let \[x_0\] be in the domain of a real-valued function \[f\]. When is \[f\] decreasing at \[x_0\]?
The function \[f(x)=x^3-3x^2+4x\], \[x\in\mathbf{R}\], is
