If f(x)=1 4x^ 2 +2x+1 , then its maximum value is : - Vidyadip

MCQ

If $\text{f}(\text{x})=\frac{1}{4\text{x}^{2}+2\text{x}+1}$, then its maximum value is :

✓
$\frac{4}{3}$
B
$\frac{2}{3}$
C
$1$
D
$\frac{3}{4}$

✓

Answer

Correct option: A.

$\frac{4}{3}$

Maximum value of $\frac{1}{4\text{x}^{2}+2\text{x}+1} =$ Minimum value of $4\text{x}^{2}+2\text{x}+1$
Now, $\text{f}(\text{x})=\text4\text{x}^{2}+2\text{x}+1$
lmplies that $\text{f}'(\text{x})=8\text{x}+2$
For a local maxima or a local minima, We must have $f\ '(x) = 0$
lmplies that $8\text{x}+2 =0$
lmplies that $8\text{x}=-2$
lmplies that $\text{x}=-14$
Now, $\text{f}\ ''(\text{x})=8$
lmplies that $\text{f}\ ''(\text{1})=8 > 0$
Therefore, $\text{x}=\frac{-1}{4}$ is a local minima.
Thus, $\frac{1}{4\text{x}^{2}+2\text{x}+1}$ is maximum at $\text{x}=\frac{-1}{4}$
lmplies that maximum value of $\frac{1}{4\text{x}^{2}+2\text{x}+1}=\frac{1}{4\Big(\frac{-1}{4}\Big)^{2}+2\Big(\frac{-1}{4}\Big)+1}$
$=\frac{16}{12}=\frac{4}{3}$

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 "M.C.Q (1 Marks)" 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

Given the function $f(x) = \frac{{{a^x} + {a^{ - x}}}}{2},\;(a > 2)$. Then $f(x + y) + f(x - y) = $

If X is brother of the son of Y's son. How is X related to Y?

Let $f(x) = 15-|x -10|;\,\,x \in R.$ then the set of all values of $x,$ at which the function, $g(x) = f(f(x))$ is not differentiable, is

If A is 3 × 4 matrix and B is matrix such that AB and BA are both defined, then B is of the type:

If $\text{f(x)}=|\text{x}-\text{a}|\ \phi\ (\text{x}),$ where $\phi(\text{x})$ is continuous function, then:

Choose the correct answer from the given four options.
If A and B are two events and $\text{A}\neq\phi,\text{B}\neq\phi,$ then:

The objective function of $\text{LPP}$ defined over the convex set attains its optimum value at.

Let $O$ be the origin and $\overline{ OA }=2 \hat{ i }+2 \hat{ j }+\hat{ k }, \overline{ OB }=\hat{ i }-2 \hat{ j }+2 \hat{ k }$ and $\overline{ OC }=\frac{1}{2}(\overline{ OB }-\lambda \overline{ OA })$ for some $\lambda>0$. If $|\overline{ OB } \times \overline{ OC }|=\frac{9}{2}$, then which of the following statements is (are) TRUE?

$(A)$ Projection of $\overline{ OC }$ on $\overline{ OA }$ is $-\frac{3}{2}$

$(B)$ Area of the triangle $OAB$ is $\frac{9}{2}$

$(C)$ Area of the triangle $ABC$ is $\frac{9}{2}$

$(D)$ The acute angle between the diagonals of the parallelogram with adjacent sides $\overline{ OA }$ and $\overline{ OC }$ is $\frac{\pi}{3}$

A and B draw two cards each, one after another, from a pack of well-shuffled pack of 52 cards. The probability that all the four cards drawn are of the same suit is

Let $\vec{b} = \hat{-i} +4\hat{j} +6\hat{k}$ and $\vec{c} = 2\hat{i} - 7\hat{j} - 10\hat{k}$. If $\vec{a}$ be a unit vector and the scalar triple product $[\vec{a}\ \ \vec{b}\ \ \vec{c}]$ has the greatest value, then $\vec{a}$ is equal to