Question

Let $f\left(x\right)=a_0+a_1{x}^2+a_2{x}^4+....+a_n{x}^{2n}$ when $0<a_0<a_1<.......<a_n$ then f(x) has

• A. No extremum
• B. Only one maximum
• C. Only one minimum
• D. Constant

Answer 1

Answer: (A)

No extremum

$f\left(x\right)=a_0+a_1{x}^2+a_2{x}^4+a_3{x}^6+....a_n{x}^{2n}$

When $0<a_0<a_1<a_2<...<a_n$

Now for critical points $f\left(x\right)=0$

$f^{\prime }\left(x\right)=2a_{1}x+4a_2{x}^3+6a_3{x}^5+....2n{a}_n{x}^{2n-1}=0$

$f^{\prime \prime }\left(x\right)=2a_1+4\times 3a_2{x}^2+6\times 5a_3{x}^4+....2n(2n-1)a_n{x}^{2n-1}$

$f^{\prime \prime }\left(x\right)$ is always positive

Hence, there will be no extremum