If a function f( x )= ax 3+ bx 2+ cx + d where a , b , c and d are integers and a 0 is such that f( sin (π/18))=0. Then the smallest possible value of f(1) is

Question

If a function f( x )= ax 3+ bx 2+ cx + d where a , b , c and d are integers and a >0 is such that f( sin (π/18))=0. Then the smallest possible value of f(1) is (A) 1 (B) 2 (C) 3 (D) 4

Tardigrade Admin · Accepted Answer

sin (π/18)= sin 10°, sin 30°=(1/2) text also sin 30°=3 sin 10°-4 sin 3 10° (1/2)=3 sin 10°-4 sin 3 10° 8 sin 3 10°+0 sin 2 10°-6 sin 10°+1=0 .....(1) f( sin 3 0°)=0 sin 3 10°+ b sin 2 10°+ c sin 10°+ d =0 ....(2) text comparing (1) text and (2) a =8, b =0, c =-6, d =1 text hence f (1)= a + b + c + d f (1)=3

Q. If a function f(x)=ax3+bx2+cx+d where a,b,c and d are integers and a>0 is such that f(sin18π​)=0. Then the smallest possible value of f(1) is

1

2

3

4

sin18π​=sin10∘,sin30∘=21​ also sin30∘=3sin10∘−4sin310∘ 21​=3sin10∘−4sin310∘ 8sin310∘+0sin210∘−6sin10∘+1=0 .....(1) f(sin30∘)=0 sin310∘+bsin210∘+csin10∘+d=0 ....(2) comparing (1) and (2) a=8,b=0,c=−6,d=1 hence f(1)=a+b+c+d f(1)=3

Q. If a function $f (x) = a x^{3} + b x^{2} + c x + d$ where $a, b, c$ and $d$ are integers and $a > 0$ is such that $f (sin \frac{π}{18}) = 0$ . Then the smallest possible value of $f (1)$ is