IIT-JEE 2001 Screening | Application of Derivatives Question 57 | Mathematics

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IIT-JEE 2001 Screening

MCQ (Single Correct Answer)

-0.5

If $$f\left( x \right) = x{e^{x\left( {1 - x} \right)}},$$ then $$f(x)$$ is

increasing on $$\left[ { - 1/2,1} \right]$$

decreasing on $$R$$

increasing on $$R$$

decreasing on $$\left[ { - 1/2,1} \right]$$

IIT-JEE 2001 Screening

MCQ (Single Correct Answer)

-0.5

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

$$-1$$

$$3$$

$$-3$$

$$1$$

IIT-JEE 2001 Screening

MCQ (Single Correct Answer)

-0.5

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

$$\left[ {0,1} \right]$$

$$\left( {0,\,1/2} \right]$$

$$\left[ {1/2,\,1} \right]$$

$$\left( {0,\,1} \right]$$

IIT-JEE 2000 Screening

MCQ (Single Correct Answer)

-0.5

Consider the following statements in $$S$$ and $$R$$
$$S:$$ $$\,\,\,$$$ Both $$\sin \,\,x$$ and $$\cos \,\,x$$ are decreasing functions in the interval $$\left( {{\pi \over 2},\pi } \right)$$
$$R:$$$$\,\,\,$$ If a differentiable function decreases in an interval $$(a, b)$$, then its derivative also decreases in $$(a, b)$$.
Which of the following is true ?

Both $$S$$ and $$R$$ are wrong

Both $$S$$ and $$R$$ are correct, but $$R$$ is not the correct explanation of $$S$$

$$S$$ is correct and $$R$$ is the correct explanation for $$S$$

$$S$$ is correct and $$R$$ is wrong

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