Evaluate limit of this integral using positive summability kernels

[psie]

Homework Statement

Let $\varphi$ be defined by $\varphi(x)=\frac{15}{16}(x^2-1)^2$ for $|x|<1$ and $\varphi(x)=0$ otherwise. Let $f$ be a function with a continuous derivative. Find the limit $$\lim_{n\to\infty}\int_{-1}^1n^2\varphi '(nx)f(x)dx.$$

Relevant Equations

Positive summability kernels, see e.g. Wikipedia.

Integrating the integral by parts, using that the antiderivative of $\varphi'(nx)$ is $\frac1{n}\varphi(nx)$, I get
$$\big[n\varphi(xn)f(x)\big]_{-1}^1-\int_{-1}^1 n\varphi(nx)f'(x)dx=0-\int_{-1}^1 n\varphi(nx)f'(x)dx.$$ I used the fact that $\varphi(n)$ and $\varphi(-n)$ both equal $0$, since $n\geq 1$.

However, I'm stuck here. This is a problem from a section on positive summability kernels, but I have been unable to verify what the kernel is in this exercise, if there is any. Appreciate any help.

 

[psie]

I solved it I think. $n\varphi(nx)$ is a positive summability kernel and the integral therefor evaluates to $-f'(0)$.

 

[haruspex]

You could try checking by plugging in $f(x)=x^r$.