Question
Download Solution PDFComprehension
The slope of the tangent to the curve y = f(x) at (x, f(x) ) is 4 for every real number x and the curve passes through the origin.
What is the nature of the curve?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFCalculation:
Given,
The slope of the tangent to the curve y = f(x) at (x, f(x)) is 4 for every real number x , and the curve passes through the origin.
The slope of the tangent is the derivative of the function, so we have:
\( f'(x) = 4 \)
Integrating f'(x) = 4 with respect to x :
\( f(x) = 4x + C \)
The curve passes through the origin, so when x = 0 , y = 0 . Substituting these values into the equation f(x) = 4x + C :
\( 0 = 4(0) + C \quad \Rightarrow \quad C = 0 \)
Therefore, the equation of the curve is:
\( f(x) = 4x \)
This is the equation of a straight line with a slope of 4, passing through the origin.
∴ The curve is a straight line with a slope of 4, passing through the origin.
Hence, the correct answer is option 1.
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