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

Question

Find the intervals in which the following functions are increasing or decreasing.
$f(x) = 2x^3 - 15x^2 + 36x + 1$

✓

Answer

$f(x) = 2x^3 - 15x^2 + 36x + 1$
$f'(x) = 6x^2 - 30x + 36$
$= 6(x^2 - 5x + 6)$
$= 6(x - 2)(x - 3)$
For $f(x)$ to be increasing, we must have
$f'(x) > 0$
$\Rightarrow 6(x - 2)(x - 3) > 0$
$\Rightarrow (x - 2)(x - 3) > 0$
$[$Since, $6 > 0, 6 (x - 2)(x - 3) > 0 \Rightarrow (x - 2)(x - 3) > 0]$
$\Rightarrow x < 2$ or $x > 3$
$\Rightarrow\text{x}\in(-\infty,2)\cup(3,\infty)$
So, $f(x)$ is increasing on $\text{x}\in(-\infty,2)\cup(3,\infty).$
For $f(x)$ to be decreasing, we must have,
$f'(x) < 0$
$\Rightarrow 6(x - 2)(x - 3) < 0$
$\Rightarrow (x - 2)(x - 3) < 0$
$[$Since, $6 > 0, 6 (x - 2)(x - 3) < 0 \Rightarrow (x - 2)(x - 3) < 0]$
$\Rightarrow 2 < x < 3$
$\Rightarrow\text{x}\in(2,3)$
So, $f(x)$ is decreasing on $\text{x}\in(2,3).$

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 "5 Marks Questions" from APPLICATION OF DERIVATIVES→APPLICATION OF DERIVATIVES — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

A furniture manufacturing company plans to make two products : chairs and tables. From its available resources which consists of 400 square feet to teak wood and 450 man hours. It is known that to make a chair requires 5 square feet of wood and 10 man-hours and yields a profit of Rs. 45, while each table uses 20 square feet of wood and 25 man-hours and yields a profit of Rs. 80. How many items of each product should be produced by the company so that the profit is maximum?

Prove that the function $\text{f(x)}=\begin{cases}\frac{\sin\text{x}}{\text{x}},&\text{x}<0\\\text{x}+1,&\text{x}\geq0\end{cases}$ is everywhere continuous.

Show that the following system of linear equations is consistent and also find solutions:x - y + z = 3
2x + y - z = 2
-x - 2y + 2z = 1

Evaluate the following integrals as limit of sum:
$\int\limits^{2}_{0}\big(\text{x}^2+2\text{x}+1\big)\text{dx}$

Differentiate $\tan^{-1}\Big(\frac{\text{x}-1}{\text{x}+1}\Big)$ with respect to $\sin^{-1}\big(3\text{x}-4\text{x}^3\big),$ if $-\frac{1}{2}<\text{x}<\frac{1}{2}$

If $\begin{bmatrix}\text{xy}&4\\\text{z}+6&\text{x}+\text{y}\end{bmatrix}=\begin{bmatrix}8&\text{w}\\0&6\end{bmatrix},$ then find the value of $x, y, z$ and $w.$

Find the relationship between 'a' and 'b' so that the function 'f' defind by $\text{f(x)}=\begin{cases}\text{ax}+1,&\text{if }\text{ x}\leq3\\\text{bx}+3,&\text{if }\text{ x}>3\end{cases}$ is continuous at x = 3.

Solve the following differential equation $(\text{x}+2\text{y}^2)\frac{\text{dy}}{\text{dx}}=\text{y},$ given that when x = 2, y = 1.

Find the intervals in which $\text{f}(\text{x})=\log(1+\text{x})-\frac{\text{x}}{1+\text{x}}$ is increasing or decreasing.

A window is in the form of a rectangle surmounted by a semi-circle. If the total perimeter of the window is 30 m, find the dimensions of the window so that maximum light is admitted.