Question

Verify the Cauchy-Schwarz Inequality and the triangle inequality for the given vectors and inner product.

p(x)=5x , q(x)= -2x^2+1, (p,q)= aobo+ a1b1+ a2b2

Required:
a. Compute (p,q)
b. Compute ||p|| and ||q||

Answer 1

To verify the Cauchy-Bunyakovsky-Schwarz Inequality, (p,q) must be less than (or equal to) ||p|| • ||q||

(1,1,1) is not equal to (-10,5)

Explanation:

a°b° + a^1b^1 + a^2b^2 < 5x (-2x^2 + 1)

Any algebra raised to the power of zero is equal to 1.

a°b° = 1 × 1 = 1

1 + ab + a^2b^2 < -10x^3 + 5x

The vectors:

(1,1,1) < (-10,5)

This verifies the Cauchy-Schwarz Inequality

Triangle Inequality states that for any triangle, the sum of the lengths of two sides must be greater than or equal to the length of the third side.