Question

Find the value of c and d so that f(x) is both continuous and differentiable.

Answer 1

Let us call
\[
g(x)=e^x + x^2 +c
\\h(x)= d x + 2
\]
we need
\[
h(0)= g(0) \\
h'(0)=g'(0)\\
h'(x) = d\\
g'(x)= e^x + 2 x

\]
Can you finish it?

Answer 2

\[
g(0) =1 +c= h(0) =c\\
c=1
\]

Answer 3

\[
g'(0) =1 =h'(0)=d\\
d=1

\]

Answer 4

There was a misprint
\[
g(0) =1 +c= h(0) =2\\
c=1
\]