Local Maxima

Q. Let $x_1,x_2,x_3$ be the critical points of the function $f\left(x\right)=3x^4+4x^3-12x^2+4$ and $f\left(x\right)$ has local maximum at $x_2$ and local minimum at $x_1,x_3$ $(x_1<x_3).$ The number of real roots of the equation $f\left(x\right)=0$ in the interval $[x_1,x_3]$ is equal to

Q.

Let $f\left(x\right)=\{\begin{array}{cc}|x^2-3x|+a,0\le x<\frac{3}{2}& \\ -2x+3x\ge \frac{3}{2}& \end{array}$ If f(x) has a local maximum at x =.

1. None of these

2. 
   

3. 
   

4. 
   

Q.

Let f(x) = cos 2$\pi$ x+x-[x] ([.] denotes the greatest integer function). Then number of points in [0, 10] in which f (x) assume its local maximum value, is

1. 9

2. 0

3. 10

4. Infinite

Q.

Let $f:[a,b]\to R$ be a function such that for
$c\epsilon (a,b),f^1\left(c\right)=f^{11}\left(c\right)=f^{111}\left(c\right)=f^{tv}\left(c\right)=f^v\left(c\right)=0$ then

1. f is necessarily a constant function

2. f has neither local maximum nor local minimum at x = c

3. f has local extremum at x=c

4. It is difficult to say whether (a) or (b)

Q.

A function f such that $f\left(a\right)=f^{\prime \prime }\left(a\right)=......f^{2n}\left(a\right)=0$ and f has a local maximum value b at x = a, if f (x) is

1. (x - a)²ⁿ⁺²

2. b - 1 - (x+1-a)²ⁿ⁺¹

3. b - (x - a)²ⁿ⁺²

4. (x - a)²ⁿ⁺² - b.

Q.

A function f(x) defined on [a, b] will have a local maximum at x = b if

[ h is a positive quantity tending to zero]

1. f(b-h) > f(b)

2. f(b-h) < f(b)

3. f(b-h) ≥ f(b)

4. f(b-h) ≤ f(b)

Q. If $f\left(x\right)={\left({\sin }^{2}x-1\right)}^n,\mathrm{then}x=\frac{\pi }{2}$ is a point of

1. local maximum, if n is odd
2. local minimum, if n is odd
3. local maximum, if n is even
4. local minimum, if n is even

Q. If $f\left(x\right)=(\begin{array}{c}3x^2+12x-1,-1\le x\le 2\\ 37-x,2<x\le 3\end{array}$, then

1. f(x) is increasing in [-1, 2]
2. f(x) is continuous in [-1, 3]
3. f (2) does not exist
4. f(x) has a maximum value at x = 2

Q. The number of integral values of $a$ for which the function $f\left(x\right)=\{\begin{array}{cc}{x}^3+1,& x\le 0\\ a^2-5a+7+x& x>0\end{array}$
has maxima at $x=0$ are

1. 0
2. 1
3. 2
4. infinite

Q. Let $f\left(x\right)={\int }_0^x(3u^2+2u+2)du,$ where x satisfies the inequality $lo{g}_2(1+\sqrt{6x-x^2-5})⩾0.$ If minimum and maximum values of f(x) are 'm' and 'M' respectively then $\frac{M+m}{4}$ is