Advertisements
Advertisements
प्रश्न
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x ?
उत्तर
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = x^4 - 4x\]
\[f'\left( x \right) = 4 x^3 - 4\]
\[ = 4\left( x^3 - 1 \right)\]
\[\text { For}f(x) \text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 4\left( x^3 - 1 \right) > 0 \]
\[ \Rightarrow x^3 - 1 > 0\]
\[ \Rightarrow x^3 > 1\]
\[ \Rightarrow x > 1\]
\[ \Rightarrow x \in \left( 1, \infty \right)\]
\[\text { So,}f(x)\text { is increasing on }\left( 1, \infty \right) . \]
\[\text { For }f(x) \text { to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow 4\left( x^3 - 1 \right) < 0\]
\[ \Rightarrow x^3 - 1 < 0\]
\[ \Rightarrow x^3 < 1\]
\[ \Rightarrow x < 1\]
\[ \Rightarrow x \in \left( - \infty , 1 \right)\]
\[\text { So,}f(x)\text { is decreasing on }\left( - \infty , 1 \right).\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
Find the intervals in which the following functions are strictly increasing or decreasing:
10 − 6x − 2x2
Find the intervals in which the following functions are strictly increasing or decreasing:
−2x3 − 9x2 − 12x + 1
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
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 \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5\] ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
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] ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4) ?
Find the intervals in which f(x) = log (1 + x) −\[\frac{x}{1 + x}\] is increasing or decreasing ?
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
Test whether the following functions are increasing or decreasing : f(x) = x3 – 6x2 + 12x – 16, x ∈ R.
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Prove that y = `(4sinθ)/(2 + cosθ) - θ` is an increasing function if `θ ∈[0, pi/2]`
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
Show that f(x) = 2x + cot–1x + `log(sqrt(1 + x^2) - x)` is increasing in R
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
The values of a for which the function f(x) = sinx – ax + b increases on R are ______.
The function f (x) = x2, for all real x, is ____________.
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
Let f: [0, 2]→R be a twice differentiable function such that f"(x) > 0, for all x ∈( 0, 2). If `phi` (x) = f(x) + f(2 – x), then `phi` is ______.
Let f(x) be a function such that; f'(x) = log1/3(log3(sinx + a)) (where a ∈ R). If f(x) is decreasing for all real values of x then the exhaustive solution set of a is ______.
If f(x) = x + cosx – a then ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
