Question

Find h and k so that f is a differentiable function for all values of x.

f(x) = x^2 - 4hx x≤8
8x + k x>8

Answer 1

Oh ok, a little bit different of a question. Again, you need to be continuous at x=8 so\[f _{-}(x=8)=f_{+}(x=8)\]\[x^{2}-4hx=8x+k \rightarrow x=8\]\[64-32h=64+k\]\[-32h=k\]Also, you need\[f_{-}^{\prime}(x=8)=f_{+}^{\prime}(x=8)\]\[2x-4h=8 \rightarrow x=8\]\[16-4h=8\]\[-4h=-8\]\[h=2\]Using this value of h you can solve for your value of k\[-32h=k \rightarrow h=2\]\[k=-64\]Hope this helps.