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1

### IIT-JEE 1997

Subjective
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.

Solve it.
2

### IIT-JEE 1996

Subjective
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.

$$\left( { - {2 \over a},{a \over 3}} \right)$$
3

### IIT-JEE 1996

Subjective
Determine the points of maxima and minima of the function
$$f\left( x \right) = {1 \over 8}\ell n\,x - bx + {x^2},x > 0,$$ where $$b \ge 0$$ is a constant.

min at $$x = {1 \over 4}\left( {b + \sqrt {{b^2} - 1} } \right)$$
max at $$x = {1 \over 4}\left( {b - \sqrt {{b^2} - 1} } \right)$$
4

### IIT-JEE 1996

Subjective
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 $$P$$.

$$y = {e^{a\left( {x - 1} \right)}}$$
Area $$=$$ $$1$$ sq. unit.

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