How to check if a function is differentiable (example from MIT)

Question

Here is a problem from MIT, we have to find the values of $a$ and $b$ which make this function differentiable:

For this function to be differentiable, it must

1) Be continuous at $x=1$

2) Have the same derivative values from both sides at $x=1$.

But the solution from MIT says:

Here, we got $14$ from $2x^5+3x^4+4x^2+5x+6$ at $x=1$, so should not it be equal to $ax^2+bx+6$ at $x=1$ instead of $a+b$?

Link of the problem and solution on MIT OpenCoursware

NOTE: Please let me know if such questions are off-topic here and I'll close it, I am fairly new to this website and am not sure how it works.

Answer 1

$2+3+4+5+6=20$.

$f$ is continuous at $1 \iff 20=a+b+6 \iff a+b=14$.

Answer 2

Let indicate

• $g(x)=ax^2+bx+6$

• $h(x)=2x^5+3x^4+4x^2+5x+6$

then we need

• for continuity $$g(1)=h(1) \implies a+b=14$$
• for differentiability $$g'(1)=h'(1)\implies 2a+b=35$$