If for the function Φ (x) = λx2 + 7x – 4, Φ’ (5) = 97, find λ.
We have to find the value of λ given in the real function and we are given with the differentiability of the function f(x) = λx2 + 7x - 4 at x = 5 which is f ‘(5) = 97, so we will adopt the same process but with a little variation.
So by using the formula, f ‘(c) = 
, we get,
f ‘(5) = 
f ‘(5) = 
f ‘(5) = 
f ‘(5) = 
as the limit has some finite value, then there must be the formation of some indeterminate form like 
, so if we put the limit value, then the numerator will also be zero as the denominator, but there must be a factor (x - 5) in the numerator, so that this form disappears.
f ‘(5) = 
f ‘(5) = limx→5 λx + 5λ + 7 = 97
f ‘(5) = 10λ + 7 = 97
10λ = 90
λ = 9
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