35:T8a9,Given that

f\left( x \right) = {x^3} + 5x + 1

29:["$","div",null,{"className":"min-w-0 max-w-[68ch] flex-1 text-[15px] leading-relaxed text-ink-600 [&_.katex-display]:my-3 [&_.katex-display]:overflow-x-auto","dangerouslySetInnerHTML":{"__html":"$35"}}] 36:T129b,

\therefore

\,\,\,\,\,

f'\left( x \right) = 3{x^2} + 5 > 0,\,\,\,\forall x \in R

2a:["$","li","1",{"className":"flex items-start gap-3","children":[["$","span",null,{"className":"mt-0.5 flex h-5 w-5 shrink-0 items-center justify-center rounded-md bg-surface-sunken font-mono text-[11px] text-ink-400","children":2}],["$","div",null,{"className":"min-w-0 max-w-[68ch] flex-1 text-[15px] leading-relaxed text-ink-600 [&_.katex-display]:my-3 [&_.katex-display]:overflow-x-auto","dangerouslySetInnerHTML":{"__html":"$36"}}]]}] 37:T698,

\Rightarrow f\left( x \right)\,\,

is strictly increasing on

R

2b:["$","li","2",{"className":"flex items-start gap-3","children":[["$","span",null,{"className":"mt-0.5 flex h-5 w-5 shrink-0 items-center justify-center rounded-md bg-surface-sunken font-mono text-[11px] text-ink-400","children":3}],["$","div",null,{"className":"min-w-0 max-w-[68ch] flex-1 text-[15px] leading-relaxed text-ink-600 [&_.katex-display]:my-3 [&_.katex-display]:overflow-x-auto","dangerouslySetInnerHTML":{"__html":"$37"}}]]}] 38:T40b,

\Rightarrow f\left( x \right)

is one one2c:["$","li","3",{"className":"flex items-start gap-3","children":[["$","span",null,{"className":"mt-0.5 flex h-5 w-5 shrink-0 items-center justify-center rounded-md bg-surface-sunken font-mono text-[11px] text-ink-400","children":4}],["$","div",null,{"className":"min-w-0 max-w-[68ch] flex-1 text-[15px] leading-relaxed text-ink-600 [&_.katex-display]:my-3 [&_.katex-display]:overflow-x-auto","dangerouslySetInnerHTML":{"__html":"$38"}}]]}] 39:T835,

\therefore

\,\,\,\,\,\,\,

Being a polynomial

f(x)

is cont. and inc.2d:["$","li","4",{"className":"flex items-start gap-3","children":[["$","span",null,{"className":"mt-0.5 flex h-5 w-5 shrink-0 items-center justify-center rounded-md bg-surface-sunken font-mono text-[11px] text-ink-400","children":5}],["$","div",null,{"className":"min-w-0 max-w-[68ch] flex-1 text-[15px] leading-relaxed text-ink-600 [&_.katex-display]:my-3 [&_.katex-display]:overflow-x-auto","dangerouslySetInnerHTML":{"__html":"$39"}}]]}] 3a:Ta9f,on

R

with

\mathop {\lim }\limits_{x \to \infty } \,f\left( x \right) = - \infty

2e:["$","li","5",{"className":"flex items-start gap-3","children":[["$","span",null,{"className":"mt-0.5 flex h-5 w-5 shrink-0 items-center justify-center rounded-md bg-surface-sunken font-mono text-[11px] text-ink-400","children":6}],["$","div",null,{"className":"min-w-0 max-w-[68ch] flex-1 text-[15px] leading-relaxed text-ink-600 [&_.katex-display]:my-3 [&_.katex-display]:overflow-x-auto","dangerouslySetInnerHTML":{"__html":"$3a"}}]]}] 3b:T877,and

\mathop {\lim }\limits_{x \to \infty } \,f\left( x \right) = \infty

2f:["$","li","6",{"className":"flex items-start gap-3","children":[["$","span",null,{"className":"mt-0.5 flex h-5 w-5 shrink-0 items-center justify-center rounded-md bg-surface-sunken font-mono text-[11px] text-ink-400","children":7}],["$","div",null,{"className":"min-w-0 max-w-[68ch] flex-1 text-[15px] leading-relaxed text-ink-600 [&_.katex-display]:my-3 [&_.katex-display]:overflow-x-auto","dangerouslySetInnerHTML":{"__html":"$3b"}}]]}] 3c:Tbfc,

\therefore

\,\,\,\,\,\,\,

Range of

f = \left( { - \infty ,\infty } \right) = R

30:["$","li","7",{"className":"flex items-start gap-3","children":[["$","span",null,{"className":"mt-0.5 flex h-5 w-5 shrink-0 items-center justify-center rounded-md bg-surface-sunken font-mono text-[11px] text-ink-400","children":8}],["$","div",null,{"className":"min-w-0 max-w-[68ch] flex-1 text-[15px] leading-relaxed text-ink-600 [&_.katex-display]:my-3 [&_.katex-display]:overflow-x-auto","dangerouslySetInnerHTML":{"__html":"$3c"}}]]}] 3d:T634,Hence

f

is onto also, So,

f

is one and onto

R.

31:["$","li","8",{"className":"flex items-start gap-3","children":[["$","span",null,{"className":"mt-0.5 flex h-5 w-5 shrink-0 items-center justify-center rounded-md bg-surface-sunken font-mono text-[11px] text-ink-400","children":9}],["$","div",null,{"className":"min-w-0 max-w-[68ch] flex-1 text-[15px] leading-relaxed text-ink-600 [&_.katex-display]:my-3 [&_.katex-display]:overflow-x-auto","dangerouslySetInnerHTML":{"__html":"$3d"}}]]}] 32:["$","footer",null,{"className":"flex flex-wrap gap-2 border-t border-rule pt-4 text-[13px] text-ink-400","children":["JEE Main 2009 · Annual, Paper 1 · Q3794",["$","span",null,{"className":"text-rule-strong","children":"·"}],"Functions"]}] 33:["$","section",null,{"className":"flex flex-col gap-3 border-t border-rule pt-6","children":[["$","h2",null,{"className":"m-0 text-[12px] font-semibold uppercase tracking-[0.06em] text-ink-400","children":["More on ","Functions"]}],["$","ul",null,{"className":"m-0 flex list-none flex-col gap-2 p-0","children":[["$","li","2a7876ab-d7f1-4c39-812a-59845e161b08",{"children":["$","$L8",null,{"href":"/q/the-domain-of-the-function-f-x-1-over-s-IuZ2hfWBOzl3se7H","className":"flex flex-col gap-0.5 rounded-xl bg-surface-card px-4 py-3 no-underline shadow-soft transition-shadow hover:shadow-lift","children":[["$","span",null,{"className":"text-[15px] font-bold text-ink-900","children":"Functions"}],["$","span",null,{"className":"text-[13px] text-ink-400","children":"JEE Main 2011 · Annual, Paper 1 · Q3841"}]]}]}],["$","li","3775d987-9975-42e7-ad22-a0dbc0e88394",{"children":["$","$L8",null,{"href":"/q/the-function-f-n-n-defined-by-f-x-igQxD7UV54aBTe7EncVNQ","className":"flex flex-col gap-0.5 rounded-xl bg-surface-card px-4 py-3 no-underline shadow-soft transition-shadow hover:shadow-lift","children":[["$","span",null,{"className":"text-[15px] font-bold text-ink-900","children":"Functions"}],["$","span",null,{"className":"text-[13px] text-ink-400","children":"JEE Main 2017 · 9 Apr, Morning shift · Q3796"}]]}]}],["$","li","f61bd8df-dd45-4f1d-a60c-e44d0290b6e9",{"children":["$","$L8",null,{"href":"/q/let-a-2b-c-1-if-f-x-left-matrix-kmmWFv4B3ZGx0zoZFq7k9k2k5khqkpe","className":"flex flex-col gap-0.5 rounded-xl bg-surface-card px-4 py-3 no-underline shadow-soft transition-shadow hover:shadow-lift","children":[["$","span",null,{"className":"text-[15px] font-bold text-ink-900","children":"Functions"}],["$","span",null,{"className":"text-[13px] text-ink-400","children":"JEE Main 2020 · 9 Jan, Evening shift · Q3863"}]]}]}],["$","li","7054cc9d-d9a4-466f-bcb7-df1b865485b5",{"children":["$","$L8",null,{"href":"/q/let-be-a-function-1ldu5gt5g","className":"flex flex-col gap-0.5 rounded-xl bg-surface-card px-4 py-3 no-underline shadow-soft transition-shadow hover:shadow-lift","children":[["$","span",null,{"className":"text-[15px] font-bold text-ink-900","children":"Functions"}],["$","span",null,{"className":"text-[13px] text-ink-400","children":"JEE Main 2023 · 25 Jan, Evening shift · Q3875"}]]}]}]]}]]}]

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