1
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.

2
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 $$P$$.
3
IIT-JEE 1996
Subjective
+5
-0
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.
4
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.

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