Calculas: finding unknowns of polynominals

• May 2nd 2008, 04:20 AM
Hibijibi
Calculas: finding unknowns of polynominals
A cubic polynomial function f is defined by f(x)= 4x^3 + ax^2 + bx + k

Where a, b, and k are constants. The function f has a local minimum at x = -1, and the graph of f has a point of inflection at x = -2.

A) Find a, and b.

B) If the integral of f(x)dx with bounds 0 to 1 is 32, what is the value of k?
• May 2nd 2008, 04:26 AM
Moo
Hello,

Quote:

Originally Posted by Hibijibi
A cubic polynomial function f is defined by f(x)= 4x^3 + ax^2 + bx + k

Where a, b, and k are constants. The function f has a local minimum at x = -1, and the graph of f has a point of inflection at x = -2.

A) Find a, and b.

B) If the integral of f(x)dx with bounds 0 to 1 is 32, what is the value of k?

A local extremum (minimum or maximum) at a point of absciss m, is defined as following :
$f'(m)=0$

Here, m=-1 and $f'(x)=\dots$

An inflection point is defined at a point of abscuss n, is defined as following :
$f''(n)=0$

Here, n=-2 and $f''(x)=\dots$
• May 2nd 2008, 04:31 AM
Isomorphism
Quote:

Originally Posted by Hibijibi
A cubic polynomial function f is defined by f(x)= 4x^3 + ax^2 + bx + k

Where a, b, and k are constants. The function f has a local minimum at x = -1, and the graph of f has a point of inflection at x = -2.

A) Find a, and b.

B) If the integral of f(x)dx with bounds 0 to 1 is 32, what is the value of k?

Use f'(-1) = 0 and f''(-2) = 0

$f''(-2) = 24(-2) + 2a = 0 \Rightarrow a = -24$

$f'(-1) = 12(-1)^2 + 2a(-1) + b = 0 \Rightarrow b = 2a - 12 = -60$

b) $\int_{0}^{1} (4x^3 - 24x^2 - 60x + k) \, dx = 32$

$(x^4 - 8x^3 - 30x^2 + kx)\bigg{|}_0^1 = 32$

$(1 - 8 - 30 + k) = 32$

$- 37 + k = 32$

$k = 69$

Hopefully I have done no computational mistakes :)