Question

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

Answer 1

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.

Answer 2

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