Past exam question on integral of an even function! I am going through a past exam question which is a function $f:\mathbb{R} \rightarrow \mathbb{R}$ is defined by
\begin{equation*}
f(x) := \int_{1}^{x^2} e^{t^2} \; dt.
\end{equation*}
The questions asked are:
1. Determine whether $f$ is even, odd or neither.
What I did was $f$ is an even function. Let $h(t) = e^{t^2}$ such that we have $\int_{1}^{x^2} h(t) \; dt$. Thus,
\begin{equation*}
\begin{split}
f(-x) &= \int_{1}^{(-x)^2} h(t) \; dt \\
&= \int_{1}^{x^2} h(t) \; dt \\
&= f(x).
\end{split}
\end{equation*}
2. If $f$ bounded below?
I know for $f$ to be bounded below there is an $m\in \mathbb{R}$ with $f(x)\geq m$, $\forall x\in \mathbb{R}$. So clearly it is because $f$ is always positive so is bounded below by $0$.
3. Explain why $f$ is differentiable and calculate its derivative $f'$.    
To find $f'$ we just differentiate both sides to get $f'(x) = e^{(x^2)^2} = e^{x^4}$ by the fundamental theorem of calculus, but how do I explain why it is differentiable?  
 A: As many of the comments have indicated, a function is even if $f(x) = f(-x)$.  This is easy to see as
\begin{equation}
f(-x) = \int_1^{(-x)^2} \exp(t^2) dt = \int_1^{x^2} \exp(t^2) dt = f(x).
\end{equation}
The function $f$ is bounded below, but it is not always positive.  It is actually bounded below by $\frac{-\sqrt{\pi}}{2}erfi(1)$, which is the value of $f(0)$.
Last, the integral of an infinitely differentiable function $h(t) = e^{t^2}$ will also be differentiable.  Using the fundamental theorem of calculus, sometimes called the Leibniz rule, the derivative is,
\begin{align}
\frac{d}{dx} f(x) &= \frac{d}{dx} \int_1^{x^2} \exp(t^2) dt\\
&= \exp((x^2)^2) \cdot \frac{d}{dx} (x^2)\\
&= 2x e^{x^4}.
\end{align}
If you need a real proof of why it is differentiable (past simply finding its derivative), then you could for example use the definition of begin differentiable, and try to show that
\begin{equation}
\lim_{h\to 0} \frac{f(x+h) - f(x)}{h}
\end{equation}
is well defined for all $x$.  In other words, you could compute the limit for general $x$, get the derivative written above, and that is sufficient proof of differentiability. 
