#### To determine

**To find:**

The values of constants a and b such that they satisfy the given equation.

**Solution: **Values of a=5, b=8

#### Explanation

**Concept:**

Apply definition of derivative of to find value of a and b.

**Formula:**

(i)

limx →0fx. gx=limx →0fx . limx →0gx

(ii) limx →af(x)-f(a)x-a=f'(a)

**Given:**

limx →0ax+b3-2x=512

**Calculations:**

Let f(x) = ax+b3

Consider limx →0ax+b3-2x, then

limx →0ax+b3-2x

= limx →0ax+b3-2x-0

= limx →0f(x)-2x-0

Compare this with definition of derivative of f(x) at 0. Then if f(0) =2, we have

512=limx →0ax+b3-2x =limx →0ax+b3-f(0)x-0=f'(0)

Now using power rule combined with chain rule

f'(x) = d/dx (ax+b)3

=13aax+b-2/3

But f'(0)=512.

Thus we have

13aa(0)+b-2/3=512

That is ab-2/3=5/4

From f(0) =2

a*0+b3=2

Thus b1/3=2

And hence b=8.

Now

ab-2/3=5/4

Substituting value of b in this we have

a8-2/3=5/4

That is a/4=5/4.

Thus it follows that a=5.

**Conclusion:**

We were able to find the values of constants a and b such that they satisfied the given equation.