Aproximation in the central difference formula

Question

I need help to solve the following problem:

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The exercise suggests following the idea of the proof of Lemma 2.4.2 which says the following:

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Following the ideas of the lemma, I try to prove the inequality given in the hint

$f(z)-f(x)=\int_{x}^{z}f'(y)dy$

$f(z)-f(x)-f'(x)(z-x)=\int_{x}^{z}[f'(y)-f'(x)]dy$

$f(z)-f(x)-f'(x)(z-x)-\frac{f''(x)(z-x)^2}{2}=\int_{x}^{z}[f'(y)-f'(x)-\frac{f''(x)(z-x)}{2}]dy$

From here I do not know how to continue, I am not sure if I should make the same change of variable as in $lemma 2.4.2$ and I do not know how to use the condition that the function $f ''$ is Lipschitz.

Can someone help me please?