Advertisements
Advertisements
प्रश्न
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?
Advertisements
उत्तर
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\]
\[ = \frac{3 x^4 + 8 x^3 - 30 x^2 - 72x + 84}{12}\]
\[f'\left( x \right) = \frac{12 x^3 + 24 x^2 - 60x - 72}{12}\]
\[ = \left( x^3 + 2 x^2 - 5x - 6 \right)\]
\[ = \left( x + 1 \right)\left( x^2 + x - 6 \right)\]
\[ = \left( x + 1 \right)\left( x - 2 \right)\left( x + 3 \right)\]
\[\text { Here }, -1, 2 \text { and } -3 \text { are the critical points }.\]
\[\text { The possible intervals are }\left( - \infty - 3 \right),\left( - 3, - 1 \right),\left( - 1, 2 \right)and\left( 2, \infty \right).\]
\[\text { For }f(x) \text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow \left( x + 1 \right)\left( x - 2 \right)\left( x + 3 \right) > 0\]
\[ \Rightarrow x \in \left( - 3, - 1 \right) \cup \left( 2, \infty \right)\]
\[\text { So },f(x)\text { is increasing on } x \in \left( - 3, - 1 \right) \cup \left( 2, \infty \right) . \]
\[\text { For }f(x) \text { to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow \left( x + 1 \right)\left( x - 2 \right)\left( x + 3 \right) < 0\]
\[ \Rightarrow x \in \left( - \infty - 3 \right) \cup \left( - 1, 2 \right) \left[ \text { From eq }. (1) \right]\]
\[\text { So,}f(x)\text { is decreasing on x } \in \left( - \infty - 3 \right) \cup \left( - 1, 2 \right) .\]
APPEARS IN
संबंधित प्रश्न
Price P for demand D is given as P = 183 +120D - 3D2 Find D for which the price is increasing
Find the intervals in which the following functions are strictly increasing or decreasing:
10 − 6x − 2x2
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.
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.
Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\] ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?
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) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Show that f(x) = sin x is an increasing function on (−π/2, π/2) ?
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).
Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R ?
If g (x) is a decreasing function on R and f(x) = tan−1 [g (x)]. State whether f(x) is increasing or decreasing on R ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
Function f(x) = ax is increasing on R, if
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
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
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
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.
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
Show that f(x) = x – cos x is increasing for all x.
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
For every value of x, the function f(x) = `1/7^x` is ______
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
The values of a for which the function f(x) = sinx – ax + b increases on R are ______.
2x3 - 6x + 5 is an increasing function, if ____________.
The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
The function f(x) = xex(1 − x), x ∈ R, is ______.
Let \[f\] be continuous on \[[a,b]\] and differentiable on \[(a,b)\]. If \[f'(x)\geq0\] for every \[x\in(a,b)\], what follows?
As one moves from left to right on a graph, what does a decrease in the \[y\]-values indicate?
