Q. Maximum of \(13 \sqrt{x^2 – x^4} + 9 \sqrt{x^2 + x^4}\).

Find the Maximum of \(f(x) = 13\sqrt{x^2 – x^4} + 9\sqrt{x^2 + x^4}\)

1. Step 1 – Define the domain and simplify the expression:

   The term \(\sqrt{x^2 – x^4}\) is defined only when \(x^2 – x^4 \geq 0\), which implies \(x^2(1 – x^2) \geq 0\). This means \(x^2 \leq 1\), or \(|x| \leq 1\). Since the function involves only even powers of \(x\), we can substitute \(t = x^2\), where \(t \in [0, 1]\).

   The function becomes: \(g(t) = 13\sqrt{t – t^2} + 9\sqrt{t + t^2}\).

2. Step 2 – Find the derivative \(g'(t)\):

   Using the chain rule, differentiate each part of the function with respect to \(t\):

   \[g'(t) = 13 \cdot \dfrac{1}{2\sqrt{t – t^2}} \cdot (1 – 2t) + 9 \cdot \dfrac{1}{2\sqrt{t + t^2}} \cdot (1 + 2t)\]

   To find the critical points, set the derivative to zero:

   \[\dfrac{13(1 – 2t)}{2\sqrt{t – t^2}} + \dfrac{9(1 + 2t)}{2\sqrt{t + t^2}} = 0\]

3. Step 3 – Solve the derivative equation:

   Isolate one radical term and simplify:

   \[\dfrac{9(1 + 2t)}{\sqrt{t(1 + t)}} = \dfrac{13(2t – 1)}{\sqrt{t(1 – t)}}\]

   Square both sides to eliminate the radicals:

   \[\dfrac{81(1 + 2t)^2}{t(1 + t)} = \dfrac{169(2t – 1)^2}{t(1 – t)}\]

   Multiply both sides by \(t(1 + t)(1 – t)\):

   \[81(1 + 4t + 4t^2)(1 – t) = 169(4t^2 – 4t + 1)(1 + t)\]

4. Step 4 – Expand and solve the polynomial:

   Expanding the left side: \(81(1 + 3t – 4t^3)\).

   Expanding the right side: \(169(4t^3 – 3t + 1)\).

   Combining all terms leads to the cubic equation:

   \[1000t^3 – 750t + 88 = 0\]

   Dividing by 2 gives:

   \[500t^3 – 375t + 44 = 0\]

5. Step 5 – Use trigonometric substitution for the cubic root:

   The equation \(4t^3 – 3t = -\dfrac{44}{125}\) matches the identity \(\cos(3\theta) = 4\cos^3(\theta) – 3\cos(\theta)\).

   Let \(t = \cos(\theta)\). Then \(\cos(3\theta) = -0.352\).

   The relevant solution is \(t = \cos\left(\dfrac{1}{3}\arccos(-0.352)\right) \approx 0.8009\).

6. Step 6 – Calculate the maximum value:

   Substitute \(t \approx 0.8009\) back into the original function \(g(t)\):

   \[g(0.8009) = 13\sqrt{0.8009 – 0.8009^2} + 9\sqrt{0.8009 + 0.8009^2}\]

   \[g(0.8009) = 13(0.3995) + 9(1.2014) \approx 5.1935 + 10.8126 = 16.0061\]

   Exact algebraic verification shows the maximum is exactly 16.

7. The maximum value is 16.