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

Question

Find the intervals in which the following functions are increasing or decreasing.
$f(x) = 6 - 9x - x^2$

✓

Answer

We have,
$f(x) = 6 - 9x - x^2$
$f'(x) = -2x - 9$
For f(x) to be increasing, we must have
$f'(x) > 0$
$⇒ -2x - 9 > 0$
$⇒ -2x > -9$
$\Rightarrow\text{x}<\frac{-9}{2}$
$\Rightarrow\text{x}\in\Big(-\infty,\frac{-9}{2}\Big)$
So, f(x) is increasing on $\Big(-\infty,\frac{-9}{2}\Big).$
For f(x) to be decreasing, we must have
$f'(x) < 0$
$⇒ -2x - 9 < 0$
$⇒ -2x < -9$
$\Rightarrow\text{x}>\frac{-9}{2}$
$\Rightarrow\text{x}\in\Big(\frac{-9}{2},\infty\Big)$
So, f(x) is decreasing on $\Big(\frac{-9}{2},\infty\Big).$

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 Increasing and Decreasing Functions→Increasing and Decreasing Functions — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

In a bank principal increases at the rate of r% par year. Find the value of r if ₹100 double it self in 10 years $(\log_e2 = 0.6931).$

If $\int\limits_{0}^{\text{k}}\frac{1}{2+8\text{x}^2}\text{ dx}=\frac{\pi}{16},$ find the value of k.

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{4}}_{-\frac{\pi}{4}}\frac{\tan^{2}\text{x}}{1+\text{e}^{\text{x}}}\text{ dx}$

Find the value of $\tan^{-1}\Big(\frac{\text{x}}{\text{y}}\Big)-\tan^{-1}\Big(\frac{\text{x-y}}{\text{x+y}}\Big)$

If the sum of lengths of the hypotenuse and a side of a right angled triangle is given, show that the area of the triangle is maximum, when the angle between them is $\frac{\pi}{3}$

Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}\cos(\text{x}+\text{y})&-\sin(\text{x}+\text{y})&\cos2\text{y}\\\sin\text{x}&\cos\text{x}&\sin\text{y}\\-\cos\text{x}&\sin\text{x}&-\cos\text{y} \end{vmatrix}$

Evaluate the following integrals as limit of sum:
$\int\limits^{\frac{\pi}{2}}_{0}\cos\text{x dx}$

Show the solution zone of the following inequalities on a graph paper:
$5\text{x}+\text{y}\geq10$
$\text{x}+\text{y}\geq6$
$\text{x}+4\text{y}\geq12$
$\text{x}\geq,\text{y}\geq0$

A telephone company in a town has $500$ subscribers on its list and collects fixed charges of Rs. 3$00/-$ per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Re $1/-$ one subscriber will discontinue the service. Find what increase will bring maximum profit$?$

$\overrightarrow{\text{n}}$ is a vector of magnitude $\sqrt{3}$ and is equally inclined to an acute angle with the coordinate axes. Find the vector and cartesian form of the equation of a plane which passes through (2, 1, -1) and is normal to $\overrightarrow{\text{n}}$