    Question

# Assertion :Let f be a real valued function satisfying f(xy)=f(x)−f(y) & limx→0f(1+x)(x)=4. Then area bounded by the curve y=f(x), the y-axis & the line y=4 is 4e square units. Reason: The function f(x) is concave downward.

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Both Assertion and Reason are incorrect
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Solution

## The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for AssertionGiven f(xy)=f(x)−f(y) ...(1)Putting x=y=1, we getf(1)=0Now f′(x)=limh→0f(x+h)−f(x)h (fact)=limh→0f(1+hx)h using (1)=limh→0f(1+hx)xhx=1xlimh→0f(1+hx)hx=4x [∵limx→0f(1+x)x=4]∴f(x)=4logx+cFor x=1;f(1)=0+c∴ c=0 as f(1)=0∴ f(x)=4logx=y (say)∴ x=ey4∴ Required area =∫4−∞xdx=∫4−∞ey4dy =4(ey4)4−∞=4(e−0)4eAgain f(x)=4logxf′(x)=4x ⇒ f′′(x)=−4x2<0∀x∈R⇒ f(x) is concave downward curveHence both assertion and reason are true but reason is not correct explanation of assertion.   Suggest Corrections  0      Similar questions  Related Videos   Line and a Point, Revisited
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