#### To determine

**To find:**

The function f

#### Answer

**12x**

#### Explanation

**1) Concept:**

Power rule of integration:

**∫abf(x)dx=xn+1n+1ab**

The Fundamental Theorem of Calculus:

Fx=ddx∫0xf(t)dt then Fx=f(x)

**2) Given:**

∫0xf(t)dt=fx2, for all x

3) **Calculation:**

It is given that f is differentiable function such that fx is never zero and

∫0xftdt=fx2

Take the derivative of both sides.

ddx∫0xftdt=ddxfx2

Since f is differentiable it is continuous so by using Fundamental Theorem of Calculus,

fx=2fxf'x

It is given that fx is never zero, so divide both sides by fx.

1=2f'x

f'x=12

Integrate on both sides with respect to x.

∫f'xdx=∫12dx

fx=12x+C, where C is constant

Substitute fx in the given equation.

∫0x12t+Cdt=12x+C2

x24+Cx=14x2+xC+C2

0=C2

C=0

So fx becomes

fx=12x

**Conclusion:**

**fx=12x**