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Navigating Media Relations As A Professional Athlete In The Spotlight Today! How To Balance Personal Branding With Press Obligations?1) Let ( f(x) = x^4 + ax^3 + bx^2 + cx + d ). Show that ( f(x) geqslant 0 ) for all ( x in mathbb{R} ) if
And only if ( a^4 + b^2 c^2 + d^4 = min(f(x)) ).
Hint:
[ f(x) = left( x^2 + frac{ax}{2} + d right)^2 + px^2 + qx + r ]
Show that ( f(x) geqslant min(f(x)) ), so ( min(f(x)) = k ).
We must have ( p = p_1^2 ), ( q = 2 p_1 q_1 ), ( r = q_1^2 + k ),
( p_1, q_1 in mathbb{R} )
Also show that when ( f(x) = (x^2)^2 ) or ( f(x) = (x^2 + d)^2 ), we have
[ k = a^4 + b^2 c^2 + d^4 ].
2) Let ( f(x) = x^n + a_1 x^{n-1} + ldots + a_n )
Given that ( f(x) geqslant 0 ), ( x in mathbb{R} ),
Show that ( g(x) = f(x) + f'(x) + ldots + f^{(n)}(x) geqslant 0 ),
Where ( f^n(x) = n! cdot a_n ).
solution: ## Part 1
To show that ( f(x) = x^4 + ax^3 + bx^2 + cx + d geqslant 0 ) for all ( x in mathbb{R} ) if and only if ( a^4 + b^2 c^2 + d^4 = min(f(x)) ), we start by considering the hint:
[ f(x) = left( x^2 + frac{ax}{2} + d right)^2 + px^2 + qx + r ]
### Step-by-Step Solution:
1. **Expand the given form:**
[
f(x) = left( x^2 + frac{ax}{2} + d right)^2 + px^2 + qx + r
]
Expanding the square term:
[
left( x^2 + frac{ax}{2} + d right)^2 = x^4 + ax^3/2 + dx^2 + (ax/2)^2/4 + adx/1 + d^2
]
Simplifying:
[
= x^4 + ax^3/2 + dx^2 + a^2x^2/4 + adx/1 + d^2
]
Combining like terms:
[
= x^4 + ax^3/2 + (d + a^2/4)x^2 + adx/1 + d^2
]
Adding ( px^2 + qx + r ):
[
f(x) = x^4 + ax^3/2 + (d + a^2/4)x^2 + adx/1 + d^2 + px^2 + qx + r
]
Grouping terms:
[
f(x) = x^4 + ax^3/2 + (d + a^2/4 + p)x^2 + (ad/1 + q)x + (d^2 + r)
]
Comparing coefficients with ( f(x) = x^4 + ax^3 + bx^2 + cx + d ):
- Coefficient of ( x^3 ): ( a/2 = a Rightarrow a = 0 )
- Coefficient of ( x^2 ): ( d + a^2/4 + p = b Rightarrow d/4+ p= b)
- Coefficient of ( x): ( ad/1+q=cRightarrow q=c)
- Constant term: ( d ^{d}+r=dRightarrow r=d-d^{d})
### Conditions for Non-negativity:
For ( f(x) = (x²+x+d²)geq0) always:
- When comparing coefficients we get:
[a=0]
[b=d+p]
[c=q]
[d=r+d-d²]
### Minimum Value:
To find minimum value:
[f'(x)=0]
[f''(x)geq0]
[f'''(x)leq0]
The minimum value occurs at:
[k=a⁴+b²c²+d⁴]
### Special Cases:
- When (f(x)=(x²)^²) or (f(x)=(x²+d²)):
[k=a⁴+b²c²+d⁴]
## Part 2
To show that if ( f(x) = x^n+ a₁x^{n-1}+...+aₙ) where (f(x)geq0) then:
[g(x)=f(x)+f'(x)+...+f^{(n)}(x)geq0]
### Step-by-Step Solution:
Given:
[f(x)=x^n+a₁x^{n-1}+...+aₙ]
### Derivatives:
[f'(x)=nx^{n-1}+(n-1)a₁x^{n-₂}+...+a_{n-1}]
[f''(x)=n(n-1)x^{n-₂}+(n-1)(n-₂)a₁x^{n-₃}+...+a_{n-₂}]
...
[f^{(n)}(x)=n!cdot aₙ]
### Summing up:
[g(x)=f(x)+f'(x)+...+f^{(n)}(x)]
Since each term is non-negative:
[g(x)geq0]itics evolved during his time?