consider the function f(x) = 12(x-1)^-2 find the value c such that the triangle formed by the tangent line to f at c and the axes has the largest possible area c=______________________ find the maximum area of the triangle formed by the tangent line to f at c and the axes A=_______________________________units^2

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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consider the function f(x) = 12(x-1)^-2

find the value c such that the triangle formed by the tangent line to f at c and the axes has the largest possible area

c=______________________

find the maximum area of the triangle formed by the tangent line to f at c and the axes

A=_______________________________units^2

Expert Solution
Step 1

Equation of tangent is: 

y-fcx-c=ddxfxx=cy-12c-12x-c=-2×12c-13yc-12-12x-c=-24c-1

Step 2

Point wherein the tangent will touch x-axis is y=0 or: 

0.c-12-12x-c=-24c-1x=c-12+cx=3c-12

Step 3

Point wherein the tangent will touch y-axis is x=0 or: 

y.c-12-120-c=-24c-1y.c-12-12=24cc-1y.c-12-12=24c-1+1c-1y.c-12-12=24+24c-1y.c-12=36+24c-1y=36c-12+24c-13y=36c-1+24c-13y=123c-1c-13

Step 4

Thus area of triangle is A=12xy or Ac=12123c-1c-133c-12 or Ac=33c-12c-13

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