Question

Determine whether the curve has a tangent at x=1. If it does, give its slope.
f(x) {x^2-2, x1

Answer 1

\[f(x) = \left\{\begin{array}{rcc}
x^2-2 & \text{if} & x \leq 1 \\
1.5x-2.5 & \text{if} & x >1
\end{array}
\right.\]

Answer 2

it is continuous at x = 1 because the limit from the left and from the right is 1

Answer 3

but it does not have a slope there. the derivative of
\[x^2-2\] is
\[2x\] and at x = 1 you get 2

Answer 4

whereas the derivative of a line is the slope, so the derivative of
\[1.5x-2.5\] is 1.5

Answer 5

and since
\[2\neq 1.5\] not differentiable at 1

Answer 6

(A)1.5 (B)2 (C)2.5 (D)3 (E)No Tangent

So the answer would be E?