First slide

Tangents and normals

Question

$If the slope of line through the origin is tangent to the curve y = x^{3} + x + 16 is m, then the value of m is$

Moderate

A

11
B

12
C

13
D

14

Solution

$T h e g i v e n c u r v e i s y = x^{3} + x + 16$
The slope of the line passing through the origin and which touches the above curve at $(x_{1}, y_{1})$ is $\frac{y_{1}}{x_{1}}$
$\frac{d y}{d x} = 3 x^{2} + 1 \Rightarrow m = 3 {x_{1}}^{2} + 1 = \frac{y_{1}}{x_{1}} \Rightarrow 3 {x_{1}}^{3} + x_{1} = {x_{1}}^{3} + x_{1} + 16 \Rightarrow x_{1} = 2 H e n c e, m = 13$

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