Advertisements
Advertisements
प्रश्न
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 7 ?
Advertisements
उत्तर
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = 2 x^3 - 24x + 7\]
\[f'\left( x \right) = 6 x^2 - 24\]
\[ = 6 \left( x^2 - 4 \right)\]
\[ = 6 \left( x + 2 \right)\left( x - 2 \right)\]
\[\text { For }f(x) \text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 6 \left( x + 2 \right)\left( x - 2 \right) > 0\]
\[ \Rightarrow \left( x + 2 \right)\left( x - 2 \right) > 0 \left[ \text { Since } 6 > 0, 6 \left( x + 2 \right)\left( x - 2 \right) > 0 \Rightarrow \left( x + 2 \right)\left( x - 2 \right) > 0 \right]\]
\[ \Rightarrow x < - 2 \ or \ x > 2\]
\[ \Rightarrow x \in \left( - \infty , - 2 \right) \cup \left( 2, \infty \right)\]
\[\text { So },f(x)\text { is increasing on } x \in \left( - \infty , - 2 \right) \cup \left( 2, \infty \right).\]
\[\text { For }f(x) \text { to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow 6 \left( x + 2 \right)\left( x - 2 \right) < 0\]
\[ \Rightarrow \left( x + 2 \right)\left( x - 2 \right) < 0 \left[ \text { Since }6 > 0, 6 \left( x + 2 \right)\left( x - 2 \right) < 0 \Rightarrow \left( x + 2 \right)\left( x - 2 \right) < 0 \right]\]
\[ \Rightarrow - 2 < x < 2\]
\[ \Rightarrow x \in \left( - 2, 2 \right)\]
\[\text { So },f(x)\text { is decreasing on }x \in \left( - 2, 2 \right) .\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the following functions are strictly increasing or decreasing:
(x + 1)3 (x − 3)3
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.
Find the intervals in which the function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 − 36x + 2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 + 12x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = (x − 1) (x − 2)2 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4)?
Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
The function f(x) = x9 + 3x7 + 64 is increasing on
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
Find `dy/dx,if e^x+e^y=e^(x-y)`
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
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 f(x) = x – cos x is increasing for all x.
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
The slope of tangent at any point (a, b) is also called as ______.
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
The function f(x) = 9 - x5 - x7 is decreasing for
The area of the square increases at the rate of 0.5 cm2/sec. The rate at which its perimeter is increasing when the side of the square is 10 cm long is ______.
Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
Find the value of x for which the function f(x)= 2x3 – 9x2 + 12x + 2 is decreasing.
Given f(x) = 2x3 – 9x2 + 12x + 2
∴ f'(x) = `squarex^2 - square + square`
∴ f'(x) = `6(x - 1)(square)`
Now f'(x) < 0
∴ 6(x – 1)(x – 2) < 0
Since ab < 0 ⇔a < 0 and b < 0 or a > 0 and b < 0
Case 1: (x – 1) < 0 and (x – 2) < 0
∴ x < `square` and x > `square`
Which is contradiction
Case 2: x – 1 and x – 2 < 0
∴ x > `square` and x < `square`
1 < `square` < 2
f(x) is decreasing if and only if x ∈ `square`
y = log x satisfies for x > 1, the inequality ______.
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 ______.
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 ______.
In the procedure for finding intervals, which conclusion is correct when \[f'(x)<0\]?
For \[f(x)=x^3-3x^2+4x\], what is \[f'(x)\]?
