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1

### IIT-JEE 2001 Screening

Let $$f\left( x \right) = \left( {1 + {b^2}} \right){x^2} + 2bx + 1$$ and let $$m(b)$$ be the minimum value of $$f(x)$$. As $$b$$ varies, the range of $$m(b)$$ is
A
$$\left[ {0,1} \right]$$
B
$$\left( {0,\,1/2} \right]$$
C
$$\left[ {1/2,\,1} \right]$$
D
$$\left( {0,\,1} \right]$$
2

### IIT-JEE 2001 Screening

The triangle formed by the tangent to the curve $$f\left( x \right) = {x^2} + bx - b$$ at the point $$(1, 1)$$ and the coordinate axex, lies in the first quadrant. If its area is $$2$$, then the value of $$b$$ is
A
$$-1$$
B
$$3$$
C
$$-3$$
D
$$1$$
3

### IIT-JEE 2001 Screening

If $$f\left( x \right) = x{e^{x\left( {1 - x} \right)}},$$ then $$f(x)$$ is
A
increasing on $$\left[ { - 1/2,1} \right]$$
B
decreasing on $$R$$
C
increasing on $$R$$
D
decreasing on $$\left[ { - 1/2,1} \right]$$
4

### IIT-JEE 2000 Screening

Let $$f\left( x \right) = \left\{ {\matrix{ {\left| x \right|,} & {for} & {0 < \left| x \right| \le 2} \cr {1,} & {for} & {x = 0} \cr } } \right.$$ then at $$x=0$$, $$f$$ has
A
a local maximum
B
no local maximum
C
a local minimum
D
no extremum

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