Find the intervals in which the following functions are increasin… - Vidyadip

Question

Find the intervals in which the following functions are increasing or decreasing. $f(x) = 6+ 12x + 3x^2 - 2x^3$

✓

Answer

$f(x) = 6+ 12x + 3x^2 - 2x^3$
$f'(x) = 12 + 6x - 6x^2$
$= -6(x^2 - x - 2)$
$= -6(x - 2)(x + 1)$
For $f(x)$ to be increasing, we must have
$f'(x) > 0$
$\Rightarrow -6(x - 2)(x + 1) > 0$
$\Rightarrow (x - 2)(x + 1) < 0$
$[$Since$, -6 < 0, -6(x - 2)(x + 1) > 0$
$\Rightarrow (x - 2)(x + 1) < 0]$
$\Rightarrow -1 < x < 2$
$\Rightarrow\text{x}\in(-1,2)$
So,$ f(x)$ is increasing on $(-1, 2).$
For $f(x)$ to be decreasing, we must have,
$f'(x) < 0$
$\Rightarrow -6(x - 2)(x + 1) < 0$
$\Rightarrow (x - 2)(x + 1) < 0$
$[$Since, $-6 < 0, -6(x - 2)(x + 1) > 0$
$\Rightarrow (x - 2)(x + 1) > 0]$
$\Rightarrow x < -1$ or $x > 2$
$\Rightarrow\text{x}\in(-\infty,-1)\cup(2,\infty)$
So, f(x) is decreasing on $(-\infty,-1)\cup(2,\infty).$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from Increasing and Decreasing Functions→Increasing and Decreasing Functions — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Show that the Modulus Function f: R → R, given by f (x) = |x|, is neither one-one nor onto, where |x| is x, if x is positive or 0 and |x| is –x, if x is negative.

A ladder, 5 metre long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides down wards at the rate of 10cm/ sec, then find the rate at which the angle between the floor and ladder is decreasing when lower end of ladder is 2 metres from the wall.

Show that the following systems of linear equations has infinite number of solutions and solve:

x - y + 3z = 6,

x + 3y - 3z = -4,

5x + 3y + 3z = 10

Find the perpendicular distence of the point (3, -1, 11) from the line $\frac{\text{x}}{2}=\frac{\text{y}-2}{-3}=\frac{\text{z}-3}{4}.$

Evaluate the following integrals:
$\int^\limits9_0\text{f(x)}\text{dx},$ Where $\text{f(x)}=\begin{cases}\sin\text{x},&0\leq\text{x}\leq\frac{\pi}{2}\\1,&\frac{\pi}{2}\leq\text{x}\leq3\\\text{e}^{\text{x}-3},&3\leq\text{x}\leq9\end{cases}$

Evaluate the following integrals:
$\int\big\{\sqrt{\text{x}}\big(\text{ax}^2+\text{bx}+\text{c}\big)\big\}\text{dx}$

If $4\sin^{-1}\text{x}+\cos^{-1}\text{x}=\pi,$ then what is the value of x?

Evalute the following integrals:
$\int\sqrt{\frac{1-\sin2\text{x}}{1+\sin2\text{x}}}\text{dx}$

Find the points o local maxima or local minima, if any, of the following functions, using the first derivatives test. Also, find the local maximum or local minimum values, as the case may be:
$\text{f}(\text{x})=\text{x}^{3}-6\text{x}^{2}+9\text{x}+15$

Find $\frac{\text{dy}}{\text{dx}}$ in the following cases:
$\frac{\text{x}^2}{\text{a}^2}+\frac{\text{y}^2}{\text{b}^2}=1$