Find the value(s) of x for which y = [x (x – 2)]^2 is an increasi… - Vidyadip

Question

Find the value(s) of $x$ for which $y = [x (x – 2)]^2 $ is an increasing function.

✓

Answer

$\text{y} = \big[\text{x}(\text{x} - 2)\big]^{2} =\big[\text{x}^{2} - 2 \text{x}\big]^{2}\therefore\frac{\text{dy}}{\text{dx}} = 2 (\text{x}^{2} - 2 \text{x})(2\text{x} - 2 )$
$\Rightarrow\frac{\text{dy}}{\text{dx}} = 4\text{x}(\text{x} - 1)(\text{x} - 2)$
$\frac{\text{dy}}{\text{dx}} = 0 \Rightarrow\text{x} = 0 , \text{x} =1 ,\text{x} = 2 $
$\therefore\text { Intervals are } ( - \infty, 0), (0,1),(1,2)(2,\infty)$
since $\frac{\text{dy}}{\text{dx}} >0\text{ in }(0,1)\text{ or } (2,\infty)$
$\therefore\text{ f(x) is increasing in}(0,1)\bigcup(2, \infty).$

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 "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

Differentiate the following functions with respect to x:
$\text{e}^{\tan^{-1}\sqrt{\text{x}}}$

If AD is the median of $\triangle\text{ABC},$ using vectors, prove that $\text{AB}^2+\text{AC}^2=2\big(\text{AD}^2+\text{CD}^2\big).$

Evaluate the following intregals:
$\int\frac{\text{x}^2+\text{x}+1}{\text{x}^2-1}\text{ dx}\int\frac{\text{x}^2+\text{x}+1}{\text{x}^2-1}\ \text{dx}$

Evaluate the following definite integral as limit of sum:$\int_\limits{0}^{4}\text{(x}+\text{e}^{2\text{x}})\ \text{dx}$

Solve the following differential equation:
$\text{x}\frac{\text{dy}}{\text{dx}}+2\text{y}=\text{x}\cos\text{x}$

Let $\overrightarrow{\text{a}} = \hat{\text{i}} + 4\hat{\text{j}} +2\hat{\text{k}}, \overrightarrow{\text{b}} = 3\hat{\text{i}} - 2\hat{\text{j}} +7\hat{\text{k}}$ and $\overrightarrow{\text{c}} = 2\hat{\text{i}} - \hat{\text{j}} + 4\hat{\text{k}}$ Find a vector $\overrightarrow{\text{d}}$ which is perpendicular to both $\overrightarrow{\text{a}} \text{and} \overrightarrow{\text{b}}\text{and} \overrightarrow{\text{c}} . \overrightarrow{\text{d}} = 27.$

Solve the following system of equations by matrix method:$\frac{2}{\text{x}}-\frac{3}{\text{y}}+\frac{3}{\text{z}}=10$
$\frac{1}{\text{x}}+\frac{1}{\text{y}}+\frac{1}{\text{z}}=10$
$\frac{3}{\text{x}}-\frac{1}{\text{y}}+\frac{2}{\text{z}}=13$

Evaluate the following integrals:$\int\frac{\text{x}+2}{2\text{x}^2+6\text{x}+5}\text{ dx}$

If $\text{A}=\begin{bmatrix}1&5\\7&12\end{bmatrix}$ and $\text{B}=\begin{bmatrix}9&1\\7&8\end{bmatrix},$ find a matrix C such that 3A + 5B + 2C is a null matrix.

$\int\frac{\text{x}-1}{\sqrt{\text{x}+4}}\ \text{dx}$