Verify Lipschitz condition of $f(t,x)=t^3e^{-tx^2}$, $(t,x)\in [0,1]\times \mathbb{R}$

Question

Verify Lipschitz condition for $f(t,x)=t^3e^{-tx^2}$, $(t,x)\in [0,1]\times \mathbb{R}$. I am having big troubles solving this problem. I'm using the formula $\frac{|f(t,x_1)-f(t,x_2)|}{|x_1-x_2|}$ and I get $t^3\cdot\frac{|e^{-tx_1^2}-e^{-tx_2^2}|}{|x_1-x_2|}$, but I don't know what else to do.

Answer 1

Hint: If you show that $|\nabla f(t,x)|$ is bounded on $[0,1]\times \mathbb R,$ then the mean value inequality gives you Lipschitz.