f(x)=x(sqrt(x^2+3))+10

Find the derivative, f'(1) and second deriv. f"(1)

Then find the value of x where the second derivative equals 0. That is, solve the equation f"(x)=0 for x.

Question

f(x)=x(sqrt(x^2+3))+10

Find the derivative, f'(1) and second deriv. f"(1)

Then find the value of x where the second derivative equals 0. That is, solve the equation f"(x)=0 for x.

campbell_st · Answer

lots of ways ot do this rewriting is the 1st step

$$f(x) = (x^4 + 3x^2)^{1/2} + 10$$

use the chain rule   the constant 10 will disappear
put the x inside the square root

$$u = x^4 + 3x^2$$        
$$du/dx = 4x^3 + 6x$$ 

then $$y = u^{1/2}$$

$$dy/du = 1/2u^{-1/2}$$

$$dy/dx = dy/du \times du/dx$$

then $$dy/dx = x \sqrt{x^2 + 3} \times (4x^3 + 6x)$$

so$$f'(x) = (4x^4 + 6x^2)\sqrt{x^2 + 3}$$

for f"(x) use the product rule... 

good luck