How do I show that a function has a unique root?

Question

My question is:

1a) Show that the function $f(x)=x^3+\sin({\pi x \over 2})−1$,for $x \in [0,1]$,has a unique root $x^* \in [0,1]$.

How would I show this using numerical methods?

b) Write down the function g for the Newton–Raphson method to compute the root $x^*$, in the form:

$x_0 \in [0,1]$ and $x_k =g(x_{k-1})$ for k≥1

Is it reasonable to expect convergence?

I'm really struggling with how to answer these questions

Thanks in advance for any help!

Answer 1

Hint: The derivative is $f’(x)=3x^2+\frac{\pi}{2}\cos{\frac{\pi}{2}x}$.

1. Justify that $f’(x)\ge 0$ over $[0,1]$. What can you deduce for $f$ (monotonicity)?

2. what is the sign of $f(0)$ and $f(1)$. What can you deduce using Intermediate Value Theorem.

3. Now use this lemma: A monotonic continuous function which changes sign over an interval $K$ vanishes a single time.