Sufficient Condition for an Extrema

Q. $f\left(x\right)=x^2-4|x|andg\left(x\right)=\{\begin{array}{c}min\left\{f\right(t):-6\le t\le x\},xϵ[-6,0]\\ max\left\{f\right(t):0\le t\le x\},xϵ[0,6]\end{array},thang\left(x\right)$

1. Exactly one point of local minima
2. Neither a point of local maxima nor minima
3. Exactly one point of local maxima
4. No point of local maxima but exactly one point of local minima

Q.

f(x) and f’(x) are differentiable at x = c. A sufficient condition for f(c) to be an extremum of f(x) is that f’(x) changes sign as x passes through c

1. True

2. False

Q. The greatest value of $f\left(x\right)=\cos (x{e}^{\left[x\right]}+7x^2-3x),x\in [-1,\infty )$ is

1. $-1$
2. $0$
3. $1$
4. None of the above.

Q. Find maximum value of y in $y=5-{(x-1)}^2$

1. None of these
2. $y_{max}=5atx=1$
3. $y_{max}=15atx=1$
4. $y_{max}=3atx=1$

Q. If $y=a\log \left|x\right|+b{x}^2+x$ has its extremum values at $x=-1$ and $x=2$ then the value of $-2ab$ is

Q. If the polynomial function $f\left(x\right)=a{x}^3+b{x}^2+cx+d$ has extreme at x=a and $x=2a$ then

1. $a,b,c$ are in AP
2. $a,b,c$ are in GP
3. $6a,4b,9c$ are in AP
4. $6a,4b,9c$ are in GP

Q. If $x=-1$ and $x=2$ are extreme points of $f\left(x\right)=\alpha \log |x|+\beta x^2+x$, then

1. $\alpha =-6,\beta =\frac{1}{2}$
2. $\alpha =-6,\beta =-\frac{1}{2}$
3. $\alpha =2,\beta =-\frac{1}{2}$
4. $\alpha =2,\beta =\frac{1}{2}$

Q. The sum of infinite terms of decreasing GP is equal to the greatest value of the function $f\left(x\right)=x^3+3x-9$ in the interval [-2, 3] and the difference between the first two terms is $f^{{}^{\prime }}\left(0\right)$. Then the common ratio of the GP is

1. $\frac{4}{3}$
2. $-\frac{2}{3}$
3. $-\frac{4}{3}$
4. $\frac{2}{3}$

Q. If greatest & least values of $f\left(x\right)={\sin }^{-1}\frac{x}{\sqrt{x^2+1}}-ln$x in $[\frac{1}{\sqrt{3}},\sqrt{3}]$ are M & m respectively, then?

1. $M+m=ln3+\frac{\pi }{6}$
2. $M=m=ln3-\frac{\pi }{6}$
3. $M+m=\frac{\pi }{2}$
4. $M-m=ln3-\frac{\pi }{3}$

Q. The maximum and minimum value of $6\sin x\cos x+4\cos 2x$ are respectively

1. $5,-5$
2. $5,5$
3. None of these
4. $-5,5$