Local Minima

Q.

If the function $f\left(x\right)=2x^3-9a{x}^2+12a^{2}x+1,$ where a > 0, attains its maximum and minimum at p and q respectively such that $p^2=q$, then a equals

1. 1

2. 2

3. 1/2

4. 3

Q.

Assuming that the petrol burnt in a motor boat varies as the cube of its velocity, the most economical speed, when going against a current of c km/h is

1. (c/2) km/h

2. (5c/2) km/h

3. (3c/4) km/h

4. (3c/2 ) km/h

Q.

$f\left(x\right)=\sin x\text{has a local minima at}x=\frac{3\pi }{2}$.

1. True

2. False

Q. Let f:$[0,1]\to R$ (the set of all real numbers) be a function. Suppose the function f is twice differentiable, $f\left(0\right)=f\left(1\right)=0$ and satisfies $f^{\prime \prime }\left(x\right)-2f^{\prime }\left(x\right)+f\left(x\right)\ge e^x,x\in [0,1]$
If the function $e^{-x}f\left(x\right)$ assumes its minimum in the interval [0, 1] at $=\frac{1}{4}$, which of the following is true?

1. $f^{\prime }\left(x\right)<f\left(x\right),\frac{1}{4}<x<\frac{3}{4}$
2. $f^{\prime }\left(x\right)<f\left(x\right),\frac{1}{4}<x<\frac{1}{4}$
3. $f^{\prime }\left(x\right)<f\left(x\right),\frac{3}{4}<x<1$
4. $f^{\prime }\left(x\right)<f\left(x\right),0<x<\frac{1}{4}$

Q. The least value of $\alpha ϵR$ for which $4\alpha x^2+\frac{1}{x}\ge 1$, for all x > 0, is

1. $\frac{1}{64}$
2. $\frac{1}{32}$
3. $\frac{1}{27}$
4. $\frac{1}{25}$

Q. The coordinates of the point on the curve $x^3=y(x-a{)}^2,a>0$ where the ordinate is minimum

1. $(2a,8a)$
2. $(-2a,\frac{-8a}{9})$
3. $(3a,\frac{27a}{4})$
4. $(-3a,\frac{-27a}{16})$

Q.

The least value of 'a' for which $\frac{4}{sinx}+\frac{1}{1-sinx}=a$ has at least one solution in the interval $(0,\pi /2)$ is

1. 1

2. 9

3. 8

4. 4

Q. The function $f\left(x\right)=2x^3–3x^2–12x+4$, has

1. two points of local maximum
2. two points of local minimum
3. one maxima and one minima
4. no maxima or minima

Q. Two towns P and Q are 60 km apart. A school is to be built to serve 150 students in town P and 50 students in town Q. If the total distance to be travelled by 200 students is minimum, then the school should be built at

1. Town P
2. Town Q
3. 45 kms from Town P
4. 45 kms from Town Q

Q. If $f\left(x\right)=x^2+\frac{1}{x^2}\text{and}g\left(x\right)=x-\frac{1}{x},xϵR-\{-1,0,1\},\text{and}h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ then the local minimum value of $h\left(x\right)$ is

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