# A cubic polynomial function f is defined by f(x)= 4x^3 + ab^2 + bx + k where a, b, and k are...

## Question:

A cubic polynomial function f is defined by {eq}f(x)= 4x^3 + ab^2 + bx + k {/eq} 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 the values of a and b.

B. If {eq}\int_{0}^{1} f(x)\ dx =32, {/eq} what is the value of K?

## Local minimum and inflection point:

In calculus, a function with one independent variable has inflection point at second derivative equal zero, and the local minimum point at first derivative equals zero.

## Answer and Explanation:

The function is:

{eq}\ f(x) = 4x^3 + ab^2 + bx + k \\ {/eq}

Its first derivative is:

{eq}\displaystyle \ f'(x)= 12x^2+b \\ {/eq}

A

The function has a local minimum at:

{eq}x=-1 {/eq} so, {eq}0= 12x^2+b \; \Rightarrow \; 0=12(-1)^2+b \; \Leftrightarrow \; b=-12 \\ {/eq}

The second derivative is:

{eq}\displaystyle \ f''(x) = 24x {/eq} so,

the function has inflection point at {eq}0=24x \; \Leftrightarrow \; x=0 {/eq}

Therefore, the function has a unique inflection point, and It is at {eq}(0,f(0)) \\ {/eq}

B

If {eq}\displaystyle \int_{0}^{1} f(x)\ dx =32 {/eq} so,

{eq}\displaystyle \int_{0}^{1} f(x)\ dx= x^4+ab^2x+ \frac{bx^2}{2}+K \bigg|_{0}^{1}=32 \\ \displaystyle (1)^4+a(-12)^2(1)+ \frac{(-12)(1)^2}{2}+K-( (0)^4+a(-12)^2(0)+ \frac{b(0)^2}{2}+K)=32 \\ \displaystyle 1+144a-6=32 \; \Leftrightarrow \; a= \frac{37}{144} \\ {/eq}

And, {eq}K \; \in \; \mathbb{R} \\ {/eq}