I have the following question here.
a) Let $f(x)=x^4-x^3+1$. Show that the graph of the function $f$ lies above the $x$-axis
Is there a way to approach this in a "nice" way? I could just sketch the graph using derivatives, concavity and such but is there a more nicer way of doing this? I feel like I could do this using integration in some way but I am not really sure.
b) Consider the region bounded by the graph of $f$, the $x$-axis, the line $x = a$, and the line $x = a + 1$. What is the value of a for which the area of this region reaches its minimum? What is the value of this minimum?
Would I simply take the bounds as $a$ and $a+1$ and then integrate the function in terms of $a$, and then maximize the function by taking the derivative with respect to $a$?
Any help would be very much appreciated!