Question

Find the values of m and b that make the following function differentiable.

Answer 1

we are given that

f(x) is differentiable at all values of x

so, f(x) will also be continuous at all values of x

f(x) will also be differentiable and continuous at x=3

f(x) is continuous at x=3:

Since, f(x) is continuous at x=3

so, the value of both function must be same at x=3

`(3)^2=m*3+b`

`3m+b=9`

f(x) is differntiable at x=3:

Since, f(x) is differentiable at x=3

so, the value of derivative of both function must be same at x=3

f'(x)=2x

f'(x)=m

now, we can set them equal at x=3

`2*3=m`

`m=6`

now, we can find b

`3*6+b=9`

`b=-9`

so,

`m=6`

`b=-9`............Answer

Answer 2

The derivative of f(x) at x=3 is 2x=6 approaching from the left side (apply power rule to y=x^2). The derivative of f(x) at x=3 is m approaching from the right side. In order for the function to be differentiable, the limit of derivative at x=3 must be the same approaching from both sides, so m=6. Then, x^2=mx+b at x=3, plug in m=6, 9=18+b, so b=-9.