Integration to find original function given the second derivative and coordinates?

Question

The curve #y = f(x)# has a stationary point at #(2, 10)# and it is given that #f''(x) = 12/x^3#. Find #f(x)#

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Answer 1 · 2017-10-12T21:01:38

See below.

If #f''(x) = 12/x^3# then #f(x) = 6/x + C_1 x+C_2#

Now we have that

#f'(2)= 0 rArr -6/2^2 + C_1=0# and also

#f(2) =6/2+2C_1+C_2 = 10#

Solving for #C_1,C_2# we obtain

#C_1 = 3/2, C_2 = 4# and finally

#f(x) = 6/x+3/2x+4#