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Q.

The coordinates of one of the points on the graph of the function $f (x) = \frac{x^{3}}{3} - \frac{5 x^{2}}{2} + 7 x - 4$ , where the tangent drawn cuts off intercepts from the coordinate axes which are equal in magnitude but opposite in sign, is

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a

$(1, \frac{5}{6})$

b

$(4, \frac{7}{2})$

c

$(2, \frac{4}{3})$

d

$(3, \frac{7}{2})$

answer is B.

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Detailed Solution

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Since the intercepts are equal in magnitude but opposite in sign ${(\frac{dy}{dx})}_{P} = 1$

$\begin{array}{l} y = f (x) = \frac{x^{3}}{3} - \frac{5 x^{2}}{2} + 7 x - 4 ∴ \frac{dy}{dx} = 1 \\ \Rightarrow x^{2} - 5 x + 7 = 1 \Rightarrow x^{2} - 5 x + 6 = 0 \Rightarrow x = 2, x = 3 ∴ x = 2 \Rightarrow y = f (2) = \frac{8}{3} \\ and x = 3 \Rightarrow y = f (3) = \frac{7}{2}; (2, \frac{8}{3}), (3, \frac{7}{2}) \end{array}$

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The coordinates of one of the points on the graph of the function f(x)=x33−5x22+7x−4, where the tangent drawn cuts off intercepts from the coordinate axes which are equal in magnitude but opposite in sign, is