Examine the function .

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Examine the function

y = (1 + 3x)^4

. (i) Compute

\dfrac{dy}{dx}

and

\dfrac{d^2y}{dx^2}

. (ii) Using the sign of

\dfrac{d^2y}{dx^2}

, explain why the graph of the function is concave upward for all real values of

x

. (iii) Describe how the result in part (ii) relates to the general behaviour of polynomial curves of even degree with positive leading coefficients.