The given function is .
f(x) is a polynomial function.
We know that a polynomial function is everywhere continuous and differentiable. So, f(x) is continuous on [−6, 6] and differentiable on (−6, 6). Thus, both the conditions of Lagrange's mean value theorem are satisfied.
So, there must exist at least one real number c ∈ (−6, 6) such that
Thus, c = 0 ∈ (−6, 6) such that .
Hence, the value of c is 0.
For the function f(x) = 8x2 − 7x + 5, x ∈ [−6, 6], the value of c for the Lagrange's mean value theorem is ___0___.