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1

### JEE Advanced 2016 Paper 2 Offline

MCQ (More than One Correct Answer)
Let f: R $$\to \left( {0,\infty } \right)$$ and g : R $$\to$$ R be twice differentiable functions such that f'' and g'' are continuous functions on R. Suppose f'$$(2)$$ $$=$$ g$$(2)=0$$, f''$$(2)$$$$\ne 0$$ and g'$$(2)$$ $$\ne 0$$. If
$$\mathop {\lim }\limits_{x \to 2} {{f\left( x \right)g\left( x \right)} \over {f'\left( x \right)g'\left( x \right)}} = 1,$$ then
A
$$f$$ has a local minimum at $$x=2$$
B
$$f$$ has a local maximum at $$x=2$$
C
$$f''(2)>f(2)$$
D
$$f(x)-f''(x)=0$$ for at least one $$x \in R$$

## Explanation

Let f : R $$\to$$ (0, $$\infty$$) and g : R $$\to$$ R.

f(x) > 0 $$\forall$$x $$\in$$ R

It is given that f'(2) = 0, g(2) = 0, f''(2) $$\ne$$ 0 and g'(2) $$\ne$$ 0.

It is also given that

$$\mathop {\lim }\limits_{x \to 2} {{f(x)g(x)} \over {f'(x)g'(x)}} = 1$$ $$\left( {{0 \over 0}} \right)$$

Applying L' Hospital rule, we get

$$\mathop {\lim }\limits_{x \to 2} {{f'(x)g(x) + g'(x)f(x)} \over {f''(x)g'(x) + f'(x)g''(x)}} = 1$$

For finite limit, we get

$${{f'(2)g(2) + g'(2)f(2)} \over {f''(2)g'(2) + f'(2)g''(2)}} = 1$$

$${{g'(2)f(2)} \over {f''(2)g'(2)}} = 1$$

$${{f(2)} \over {f''(2)}} = 1 \Rightarrow f''(2) = f(2) > 0$$ and f'(2) = 0 which means that f(x) has local minima at x = 2.

Hence, option (A) is correct.

$$f(2) - f''(2) = 0$$

Therefore, we can say that $$f(x) - f''(x) = 0$$ has at least one solution in x $$\in$$ R.

Hence, option (D) is correct.

2

### JEE Advanced 2015 Paper 2 Offline

MCQ (More than One Correct Answer)
Let $$f, g :$$ $$\left[ { - 1,2} \right] \to R$$ be continuous functions which are twice differentiable on the interval $$(-1, 2)$$. Let the values of f and g at the points $$-1, 0$$ and $$2$$ be as given in the following table:
X = -1 X = 0 X = 2
f(x) 3 6 0
g(x) 0 1 -1

In each of the intervals $$(-1, 0)$$ and $$(0, 2)$$ the function $$(f-3g)''$$ never vanishes. Then the correct statement(s) is (are)

A
$$f'\left( x \right) - 3g'\left( x \right) = 0$$ has exactly three solutions in $$\left( { - 1,0} \right) \cup \left( {0,2} \right)$$
B
$$f'\left( x \right) - 3g'\left( x \right) = 0$$ has exactly one solution in $$(-1, 0)$$
C
$$f'\left( x \right) - 3g'\left( x \right) = 0$$ has exactly one solution in $$(0, 2)$$
D
$$f'\left( x \right) - 3g'\left( x \right) = 0$$ has exactly two solutions in $$(-1, 0)$$ and exactly two solutions in $$(0, 2)$$

## Explanation

Let $$F(x) = f(x) - 3g(x)$$

$$\therefore$$ $$F( - 1) = 3$$, $$F(0) = 3$$ and $$F(2) = 3$$

So, $$F'(x)$$ will vanish at least twice in

$$( - 1,0) \cup (0,2)$$.

$$\because$$ $$F''(x) > 0$$ or $$< 0$$, $$\forall x \in ( - 1,0) \cup (0,2)$$

Hence, $$f'(x) - 3g'(x) = 0$$ has exactly one solution in $$( - 1,0)$$ and one solution in $$(0,2)$$.

3

### JEE Advanced 2013 Paper 2 Offline

MCQ (More than One Correct Answer)

The function $$f(x) = 2\left| x \right| + \left| {x + 2} \right| - \left| {\left| {x + 2} \right| - 2\left| x \right|} \right|$$ has a local minimum or a local maximum at x =

A
$$-$$2
B
$${{ - 2} \over 3}$$
C
2
D
$${{ 2} \over 3}$$

## Explanation

Here, $$f(x) = 2\left| x \right| + \left| {x + 2} \right| - \left| {\left| {x + 2} \right| - 2\left| x \right|} \right|$$

$$= \left\{ {\matrix{ { - 2x - (x + 2) + (x - 2),} & {when\,x \le - 2} \cr { - 2x + x + 2 + 3x + 2,} & {when\, - 2 < x \le - {2 \over 3}} \cr { - 4x,} & {when\, - {2 \over 3} < x \le 0} \cr {4x,} & {when\,0 < x \le 2} \cr {2x + 4,} & {when\,x > 2} \cr } } \right.$$

$$= \left\{ {\matrix{ { - 2x - 4,} & {x \le - 2} \cr {2x + 4,} & { - 2 < x \le - 2/3} \cr { - 4x,} & { - {2 \over 3} < x \le 0} \cr {4x,} & {0 < x \le 2} \cr {2x + 4,} & {x > 2} \cr } } \right.$$

Graph for y = f(x) is shown as

4

### JEE Advanced 2013 Paper 1 Offline

MCQ (More than One Correct Answer)
A rectangular sheet of fixed perimeter with sides having their lengths in the ratio $$8:15$$ is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is $$100$$, the resulting box has maximum volume. Then the lengths of the vsides of the rectangular sheet are
A
$$24$$
B
$$32$$
C
$$45$$
D
$$60$$

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