1

IIT-JEE 1998

Subjective

+8

-0

Suppose $$f(x)$$ is a function satisfying the following conditions

(a) $$f(0)=2,f(1)=1$$,

(b) $$f$$has a minimum value at $$x=5/2$$, and

(c) for all $$x$$, $$$f'\left( x \right) = \matrix{ {2ax} & {2ax - 1} & {2ax + b + 1} \cr b & {b + 1} & { - 1} \cr {2\left( {ax + b} \right)} & {2ax + 2b + 1} & {2ax + b} \cr } $$$

where $$a,b$$ are some constants. Determine the constants $$a, b$$ and the function $$f(x)$$.

(a) $$f(0)=2,f(1)=1$$,

(b) $$f$$has a minimum value at $$x=5/2$$, and

(c) for all $$x$$, $$$f'\left( x \right) = \matrix{ {2ax} & {2ax - 1} & {2ax + b + 1} \cr b & {b + 1} & { - 1} \cr {2\left( {ax + b} \right)} & {2ax + 2b + 1} & {2ax + b} \cr } $$$

where $$a,b$$ are some constants. Determine the constants $$a, b$$ and the function $$f(x)$$.

2

IIT-JEE 1997

Subjective

+5

-0

Let $$a+b=4$$, where $$a<2,$$ and let $$g(x)$$ be a differentiable function.

If $${{dg} \over {dx}} > 0$$ for all $$x$$, prove that $$\int_0^a {g\left( x \right)dx + \int_0^b {g\left( x \right)dx} } $$

increases as $$(b-a)$$ increases.

3

IIT-JEE 1996

Subjective

+3

-0

Let $$f\left( x \right) = \left\{ {\matrix{
{x{e^{ax}},\,\,\,\,\,\,\,x \le 0} \cr
{x + a{x^2} - {x^3},\,x > 0} \cr
} } \right.$$

Where a is a positive constant. Find the interval in which $$f'(x)$$ is increasing.

4

IIT-JEE 1996

Subjective

+5

-0

A curve $$y=f(x)$$ passes through the point $$P(1, 1)$$. The normal to the curve at $$P$$ is $$a(y-1)+(x-1)=0$$. If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, determine the equation of the curve. Also obtain the area bounded by the $$y$$-axis, the curve and the normal to the curve at $$P$$.

Questions Asked from Application of Derivatives (Subjective)

Number in Brackets after Paper Indicates No. of Questions

IIT-JEE 2006 (1)
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