IB AA HL May 2026 P3 TZC Q1

Consider \(f(x)=x^4+bx^2+d\), where \(b\in\mathbb{R}\) and \(d\in\mathbb{R}\), \(d\ne0\).

1. Consider the case where \(b=1\) and \(d=1\).
   1. Sketch the curve \(y=x^4+x^2+1\), labelling any axes intercepts.[2]
   2. Hence, state why the equation \(x^4+x^2+1=0\) has no real roots.[1]
2. Consider the case where \(b=-2\) and \(d=1\).
   Sketch the curve \(y=x^4-2x^2+1\), labelling any axes intercepts.[2]

3. 1. Write down an expression for \(f'(x)\).

      [2]

   2. Show that \(f'\) is an odd function.

      [3]

4. A root \(\alpha\) is a double root of \(f(x)=0\) if both \(f(\alpha)=0\) and \(f'(\alpha)=0\).

   State a feature of the graph of \(y=f(x)\) that indicates whether \(f(x)=0\) has a double root.[1]

5. In parts (e) to (g), suppose that \(f(x)=0\) has a double root \(\alpha\).

   Show that \(-\alpha\) is also a double root of \(f(x)=0\).[4]

6. 1. By considering \(f'(\alpha)=0\), or otherwise, show that \(\alpha^2=-\frac b2\).

      [3]
   2. Hence, show that \(d=\frac{b^2}{4}\).[2]
7. Deduce the set of values of \(b\) for which \(f(x)=0\) has
   1. real roots;[1]
   2. purely imaginary roots.[1]

8. In parts (h) and (i), consider \(f(x)=x^4+bx^2+d\), where \(b<0\) and \(d\ne0\).

   Show that the solutions of \(f''(x)=0\) are \(x=\pm\sqrt{-\frac b6}\).

   [3]
9. The curve \(y=f(x)\) has a point of inflexion at \(x=\sqrt{-\frac b6}\).
   The tangent there has equation \(y=-\frac{2\sqrt6}{9}(-b)^{3/2}x+c\).
   Show that \(c=\frac{b^2}{12}+d\).[5]