First Derivative Test for Local Maximum

Q. find the range of the function f(x)=sin^2x+cos^4x

Q. 65.FIND THE PERIOD OF f(x) = [sin 3x] + |cos 6x|

Q. The equation of the tangent to the hyperbola {3x^2-4y^2=12, which makes equal intercepts on the axes is

Q. 11. y-intercept of the common tangent to the parabolay2-32x and x2-108y is(1) 18(3) 9(2) 12(4) 6

Q. 39. y= Sin x -1/3sin x -2 find domain and range.

Q. 14. How to make graph of sin|x|

Q. Let $P\left(x\right)=a_0+a_1{x}^2+a_2{x}^4+......+a_n{x}^{2n}$ be a polynomial in a variable x with $0<a_0<a_1<a_2<a_3.......<a_n$. The function P(x) has

1. neither a maximum nor a minimum
2. only one maximum
3. only one minimum
4. only one maximum and only one minimum
5. none of these

Q. Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is:

1. $12.5$
2. $10$
3. $25$
4. $30$

Q. 3^x + x³ + 3³

Q. Prove the following identities (1-16)
$\frac{\cos x}{1-\sin x}=\frac{1+\cos \mathit{}x+\sin x}{1+\cos x-\sin x}$

Q. 6.A sin(wt-kx) Find dy/dx.

Q. $f\left(x\right)=\sin x\left(1+\cos x\right),x\in \left(0,\frac{\pi }{2}\right)$Then, f(x) has a maxima at .

1. $\frac{\pi }{2}$
2. $\frac{\pi }{6}$
3. $\frac{\pi }{4}$
4. $\frac{\pi }{3}$

Q. (x sin x + cos x) (x cos x − sin x)

Q. The absolute difference between the greatest and the least values of the function $f\left(x\right)=x(\ln x-2)$ on $[1,e^2]$ is

1. $2$
2. $e$
3. $e^2$
4. $1$

Q. $\underset{x\to 0}{\lim }\frac{x\cos x+2\sin x}{x^2+\tan x}$

Q. $\underset{x\to 0}{\lim }{\left(\cos x+\sin x\right)}^{1/x}$

Q. If sin x = -cos x, x is greater than 0 and less than 2pi find the possible values of x.

Q. 2. Let f(x)= tan x, x (-/2, /2) & g(x)= (3-x.x).find gof

Q. 18. If A and B are roots of acosx - bsinx=c then find tan(A+B) and tan(A+B)

Q. Q18)
prove that : sin (n+1) x sin (n+2)x + cos (n+1) x cos (n+2)x =cosx

18. Prove that : sin ( n + 1 ) x sin ( n + 2 ) x + cos ( n + 1 ) x cos ( n+ 2) x = cosx

Q.

The function $f\left(x\right)=x^2(x-2{)}^2$

1. Decreases on (0, 1) ∪ (2, $\infty$)

2. Increase on ($-\infty$, 0) $\cup$ (1, 2)

3. Has a local maximum value 0

4. Has local maximum value 1

Q. If $A=\{x\in R:|x-2|>1\},B=\{x\in R:\sqrt{x^2-3}>1\},$ $C=\{x\in R:|x-4|\ge 2\}$ and $Z$ is the set of all integers, then the number of subsets of the set $(A\cap B\cap C{)}^c\cap Z$ is

Q. Statement-1: $e^{\pi }>{\pi }^e$
Statement-2: ${\pi }^{e\pi }>e^{e\pi }$

1. Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation of Statement-1
2. Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation of Statement-1
3. Statement-1 is true, Statement-2 is false
4. Statement-1 is false, Statement-2 is true

Q. The function $f\left(x\right)=4x^5-25x^4+40x^3-10$ has

1. two local maximum and two local minimum
2. one local maximum and one local minimum
3. one local maximum and two local minimum
4. two local maximum and one local minimum

Q. $\underset{x\to 0}{\lim }\frac{\cos 2x-1}{\cos x-1}$

Q.

find the derivative of sinx+xcosx the whole divided by sinx-cosx with respect to x

Q. The value of $\underset{x\to 0}{\lim }\frac{1-\cos x+2\sin x-{\sin }^{3}x-x^2+3x^4}{{\tan }^{3}x-6{\sin }^{2}x+x-5x^3}$ is

(a) 1

(b) 2

(c) −1

(d) −2

Q. Limit (x --> 0) (sin 2x + sin 6x)/(sin 5x - sin 3x)

Q. Prove that: sin​α + sin (α + 2π/3) + sin (α + 4π/3) = 0
Kindly proceed by using sin (A+B) = sinAcosB + cosAsinB (in sin (α + 2π/3) and sin (α + 4π/3)) in first step. :)

Q. Find the integration of sin2x dx from 0 to pi/2.